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Theorem 2sb5nd 39301
Description: Equivalence for double substitution 2sb5 2276 without distinct 𝑥, 𝑦 requirement. 2sb5nd 39301 is derived from 2sb5ndVD 39668. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2sb5nd ((¬ ∀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 2sb5nd
StepHypRef Expression
1 ax6e2ndeq 39300 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 anabs5 642 . . . 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 39293 . . . . . . . . 9 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
43exbii 1924 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
5 nfs1v 2274 . . . . . . . . . 10 𝑦[𝑣 / 𝑦]𝜑
65nfsb 2590 . . . . . . . . 9 𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
7619.41 2259 . . . . . . . 8 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
84, 7bitr3i 266 . . . . . . 7 (∃𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
98exbii 1924 . . . . . 6 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
10 nfs1v 2274 . . . . . . 7 𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
111019.41 2259 . . . . . 6 (∃𝑥(∃𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
129, 11bitr2i 265 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
1312anbi2i 609 . . . 4 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
142, 13bitr3i 266 . . 3 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
15 pm5.32 563 . . 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))))
1614, 15mpbir 221 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
171, 16sylbi 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  wa 382  wo 834  wal 1629   = wceq 1631  wex 1852  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ne 2944  df-v 3353
This theorem is referenced by:  2uasbanh  39302  2uasbanhVD  39669
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