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Theorem cbvrexcsf 3707
Description: A more general version of cbvrexf 3305 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 1933 . . . 4 𝑦 ¬ 𝜑
5 cbvralcsf.4 . . . . 5 𝑥𝜓
65nfn 1933 . . . 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 3706 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓)
1110notbii 309 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
12 dfrex2 3134 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
13 dfrex2 3134 . 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 1632  wnf 1857  wnfc 2889  wral 3050  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-sbc 3577  df-csb 3675
This theorem is referenced by:  cbvrexv2  3711
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