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Theorem 2sb5ndALT 39820
Description: Equivalence for double substitution 2sb5 2289 without distinct 𝑥, 𝑦 requirement. 2sb5nd 39438 is derived from 2sb5ndVD 39798. 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 39798. (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 39437 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 anabs5 653 . . . 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 39430 . . . . . . . . 9 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
43exbii 1943 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
5 hbs1 2288 . . . . . . . . . . . 12 ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
6 id 22 . . . . . . . . . . . . 13 (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
7 axc11 2410 . . . . . . . . . . . . 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 pm3.33 781 . . . . . . . . . . . 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 581 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
11 hbs1 2288 . . . . . . . . . . . . . 14 ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
1211sbt 2510 . . . . . . . . . . . . 13 [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
13 sbi1 2483 . . . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . 15 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
1716con3i 151 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1815, 17syl 17 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
19 sbal2 2553 . . . . . . . . . . . . 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 339 . . . . . . . . . . . . 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 470 . . . . . . . . . . . 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 581 . . . . . . . . . . 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 2188 . . . . . . . . 9 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
262519.41 2268 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
274, 26bitr3i 268 . . . . . . 7 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2827exbii 1943 . . . . . 6 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
29 nfs1v 2287 . . . . . . 7 𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
302919.41 2268 . . . . . 6 (∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
3128, 30bitr2i 267 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
3231anbi2i 616 . . . 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 268 . . 3 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
34 pm5.32 569 . . 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 222 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
361, 35sylbi 208 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 197  wa 384  wo 873  wal 1650   = wceq 1652  wex 1874  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-ne 2938  df-v 3352
This theorem is referenced by: (None)
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