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Theorem cbvalv1 2174
Description: Version of cbval 2270 with a dv condition, which does not require ax-13 2245. See cbvalvw 1966 for a version with two dv conditions, requiring fewer axioms, and cbvalv 2272 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
cbvalv1.nf1 𝑦𝜑
cbvalv1.nf2 𝑥𝜓
cbvalv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalv1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvalv1
StepHypRef Expression
1 cbvalv1.nf1 . . 3 𝑦𝜑
2 cbvalv1.nf2 . . 3 𝑥𝜓
3 cbvalv1.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 219 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3v 2169 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63biimprd 238 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76equcoms 1944 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3v 2169 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 199 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wnf 1705
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:  cbvexv1  2175  cleqh  2721  bj-cbvalvv  32428  bj-cbval2v  32432  bj-abbi  32471
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