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Theorem 2sb5ndALT 38678
Description: Equivalence for double substitution 2sb5 2442 without distinct 𝑥, 𝑦 requirement. 2sb5nd 38285 is derived from 2sb5ndVD 38656. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in 2sb5ndVD 38656. (Contributed by Alan Sare, 19-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2sb5ndALT ((¬ ∀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 2sb5ndALT
StepHypRef Expression
1 ax6e2ndeq 38284 . 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 38277 . . . . . . . . 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 . . . . . . . . . . . 12 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
6 id 22 . . . . . . . . . . . . 13 (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
7 axc11 2313 . . . . . . . . . . . . 13 (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
86, 7syl 17 . . . . . . . . . . . 12 (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
9 pm3.33 608 . . . . . . . . . . . 12 ((([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, 9sylancr 694 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
11 hbs1 2435 . . . . . . . . . . . . . 14 ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
1211sbt 2418 . . . . . . . . . . . . 13 [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
13 sbi1 2391 . . . . . . . . . . . . 13 ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑))
1412, 13ax-mp 5 . . . . . . . . . . . 12 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑)
15 id 22 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
16 axc11n 2306 . . . . . . . . . . . . . . 15 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
1716con3i 150 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1815, 17syl 17 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
19 sbal2 2460 . . . . . . . . . . . . 13 (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2018, 19syl 17 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
21 imbi2 338 . . . . . . . . . . . . 13 (([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)))
2221biimpac 503 . . . . . . . . . . . 12 ((([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))
2314, 20, 22sylancr 694 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2410, 23pm2.61i 176 . . . . . . . . . 10 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2524nf5i 2021 . . . . . . . . 9 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
262519.41 2101 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
274, 26bitr3i 266 . . . . . . 7 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2827exbii 1771 . . . . . 6 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
29 nfs1v 2436 . . . . . . 7 𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
302919.41 2101 . . . . . 6 (∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
3128, 30bitr2i 265 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
3231anbi2i 729 . . . 4 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
332, 32bitr3i 266 . . 3 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
34 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))))
3533, 34mpbir 221 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
361, 35sylbi 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 3191
This theorem is referenced by: (None)
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