Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2sb5ndVD Structured version   Visualization version   GIF version

Theorem 2sb5ndVD 38626
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 38264) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. 2sb5nd 38255 is 2sb5ndVD 38626 without virtual deductions and was automatically derived from 2sb5ndVD 38626. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 2:1: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 3:: ⊢ ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) 4:3: ⊢ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) 5:4: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥] ∀𝑦[𝑣 / 𝑦]𝜑) 6:: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑥𝑥 = 𝑦   ) 7:: ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑥𝑥 = 𝑦) 8:7: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑦𝑦 = 𝑥) 9:6,8: ⊢ (   ¬ ∀𝑥𝑥 = 𝑦   ▶   ¬ ∀𝑦𝑦 = 𝑥   ) 10:9: ⊢ ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀ 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 11:5,10: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 12:11: ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 13:: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 14:: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ∀𝑥𝑥 = 𝑦   ) 15:14: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   (∀𝑥[𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 16:13,15: ⊢ (   ∀𝑥𝑥 = 𝑦   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦 ]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ) 17:16: ⊢ (∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦] 𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 19:12,17: ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) 20:19: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 21:2,20: ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 22:21: ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 23:13: ⊢ (∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 24:22,23: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) 240:24: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 241:: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ( ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) 242:241,240: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 243:: ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ( [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) ↔ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))) 25:242,243: ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) 26:: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥 ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) qed:25,26: ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
Assertion
Ref Expression
2sb5ndVD ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem 2sb5ndVD
StepHypRef Expression
1 ax6e2ndeq 38254 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 anabs5 850 . . . 4 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
3 2pm13.193 38247 . . . . . . . . 9 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
43exbii 1771 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
5 hbs1 2435 . . . . . . . . . . . . 13 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
6 idn1 38269 . . . . . . . . . . . . . 14 (   𝑥 𝑥 = 𝑦   ▶   𝑥 𝑥 = 𝑦   )
7 axc11 2313 . . . . . . . . . . . . . 14 (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
86, 7e1a 38331 . . . . . . . . . . . . 13 (   𝑥 𝑥 = 𝑦   ▶   (∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
9 imim1 83 . . . . . . . . . . . . 13 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
105, 8, 9e01 38395 . . . . . . . . . . . 12 (   𝑥 𝑥 = 𝑦   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
1110in1 38266 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
12 hbs1 2435 . . . . . . . . . . . . . . 15 ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
1312sbt 2418 . . . . . . . . . . . . . 14 [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
14 sbi1 2391 . . . . . . . . . . . . . 14 ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑))
1513, 14e0a 38478 . . . . . . . . . . . . 13 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑)
16 idn1 38269 . . . . . . . . . . . . . . 15 (    ¬ ∀𝑥 𝑥 = 𝑦   ▶    ¬ ∀𝑥 𝑥 = 𝑦   )
17 axc11n 2306 . . . . . . . . . . . . . . . 16 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
1817con3i 150 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1916, 18e1a 38331 . . . . . . . . . . . . . 14 (    ¬ ∀𝑥 𝑥 = 𝑦   ▶    ¬ ∀𝑦 𝑦 = 𝑥   )
20 sbal2 2460 . . . . . . . . . . . . . 14 (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2119, 20e1a 38331 . . . . . . . . . . . . 13 (    ¬ ∀𝑥 𝑥 = 𝑦   ▶   ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
22 imbi2 338 . . . . . . . . . . . . . 14 (([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
2322biimpcd 239 . . . . . . . . . . . . 13 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑) → (([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
2415, 21, 23e01 38395 . . . . . . . . . . . 12 (    ¬ ∀𝑥 𝑥 = 𝑦   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2524in1 38266 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2611, 25pm2.61i 176 . . . . . . . . . 10 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2726nf5i 2021 . . . . . . . . 9 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
282719.41 2101 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
294, 28bitr3i 266 . . . . . . 7 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
3029exbii 1771 . . . . . 6 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
315nf5i 2021 . . . . . . 7 𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
323119.41 2101 . . . . . 6 (∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
3330, 32bitr2i 265 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
3433anbi2i 729 . . . 4 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
352, 34bitr3i 266 . . 3 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
36 pm5.32 667 . . 3 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))) ↔ ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))))
3735, 36mpbir 221 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
381, 37sylbi 207 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701  [wsb 1877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-v 3188  df-vd1 38265 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator