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Theorem cbvrexcsf 3934
Description: A more general version of cbvrexf 3351 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2365. (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 1852 . . . 4 𝑦 ¬ 𝜑
5 cbvralcsf.4 . . . . 5 𝑥𝜓
65nfn 1852 . . . 4 𝑥 ¬ 𝜓
7 cbvralcsf.5 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
8 cbvralcsf.6 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
98notbid 318 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
101, 2, 4, 6, 7, 9cbvralcsf 3933 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓)
1110notbii 320 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
12 dfrex2 3067 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
13 dfrex2 3067 . 2 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
1411, 12, 133bitr4i 303 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wnf 1777  wnfc 2877  wral 3055  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-sbc 3773  df-csb 3889
This theorem is referenced by:  cbvrexv2  3938
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