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Theorem bj-cbvalvv 32428
Description: Version of cbvalv 2272 with a dv condition, which does not require ax-13 2245. UPDATE: this is cbvalvw 1966 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-cbvalvv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbvalvv (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bj-cbvalvv
StepHypRef Expression
1 nfv 1840 . 2 𝑦𝜑
2 nfv 1840 . 2 𝑥𝜓
3 bj-cbvalvv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvalv1 2174 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-zfpow  32491  bj-nfcjust  32550
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