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Theorem cbvexv 2426
 Description: Rule used to change bound variables, using implicit substitution. See cbvexvw 2144 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2192. (Revised by Wolf Lammen, 17-Jul-2021.)
Hypothesis
Ref Expression
cbvalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexv (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvexv
StepHypRef Expression
1 cbvalv.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
21notbid 310 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32cbvalv 2425 . . 3 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
4 alnex 1880 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5 alnex 1880 . . 3 (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
63, 4, 53bitr3i 293 . 2 (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
76con4bii 313 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1654  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-11 2207  ax-12 2220  ax-13 2389 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-nf 1883 This theorem is referenced by:  cbvex2v  2434
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