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Theorem cbvrexcsf 3873
 Description: A more general version of cbvrexf 3387 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvralcsf.1 𝑦𝐴
cbvralcsf.2 𝑥𝐵
cbvralcsf.3 𝑦𝜑
cbvralcsf.4 𝑥𝜓
cbvralcsf.5 (𝑥 = 𝑦𝐴 = 𝐵)
cbvralcsf.6 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexcsf (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)

Proof of Theorem cbvrexcsf
StepHypRef Expression
1 cbvralcsf.1 . . . 4 𝑦𝐴
2 cbvralcsf.2 . . . 4 𝑥𝐵
3 cbvralcsf.3 . . . . 5 𝑦𝜑
43nfn 1858 . . . 4 𝑦 ¬ 𝜑
5 cbvralcsf.4 . . . . 5 𝑥𝜓
65nfn 1858 . . . 4 𝑥 ¬ 𝜓
7 cbvralcsf.5 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
8 cbvralcsf.6 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
98notbid 321 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
101, 2, 4, 6, 7, 9cbvralcsf 3872 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓)
1110notbii 323 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
12 dfrex2 3202 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
13 dfrex2 3202 . 2 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
1411, 12, 133bitr4i 306 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538  Ⅎwnf 1785  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-sbc 3723  df-csb 3831 This theorem is referenced by:  cbvrexv2  3877
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