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Theorem bj-cbvaldv 33065
Description: Version of cbvald 2435 with a dv condition, which does not require ax-13 2407. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbvaldv.1 𝑦𝜑
bj-cbvaldv.2 (𝜑 → Ⅎ𝑦𝜓)
bj-cbvaldv.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-cbvaldv (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem bj-cbvaldv
StepHypRef Expression
1 nfv 1994 . 2 𝑥𝜑
2 bj-cbvaldv.1 . 2 𝑦𝜑
3 bj-cbvaldv.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfv 1994 . . 3 𝑥𝜒
54a1i 11 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 bj-cbvaldv.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
71, 2, 3, 5, 6bj-cbv2v 33062 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ex 1852  df-nf 1857
This theorem is referenced by:  bj-cbvexdv  33066  bj-cbvaldvav  33071
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