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Theorem cbvrexcsf 3599
Description: A more general version of cbvrexf 3196 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
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 1824 . . . 4 𝑦 ¬ 𝜑
5 cbvralcsf.4 . . . . 5 𝑥𝜓
65nfn 1824 . . . 4 𝑥 ¬ 𝜓
7 cbvralcsf.5 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
8 cbvralcsf.6 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
98notbid 307 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
101, 2, 4, 6, 7, 9cbvralcsf 3598 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓)
1110notbii 309 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
12 dfrex2 3025 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
13 dfrex2 3025 . 2 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
1411, 12, 133bitr4i 292 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  wnf 1748  wnfc 2780  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-sbc 3469  df-csb 3567
This theorem is referenced by:  cbvrexv2  3603
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