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Theorem cbvaldw 2351
Description: Version of cbvald 2421 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvaldw.1 𝑦𝜑
cbvaldw.2 (𝜑 → Ⅎ𝑦𝜓)
cbvaldw.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvaldw (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvaldw
StepHypRef Expression
1 nfv 1908 . 2 𝑥𝜑
2 cbvaldw.1 . 2 𝑦𝜑
3 cbvaldw.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfvd 1909 . 2 (𝜑 → Ⅎ𝑥𝜒)
5 cbvaldw.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 3, 4, 5cbv2w 2350 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1528  wnf 1777
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 1904  ax-6 1963  ax-7 2008  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by:  cbvexdw  2352
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