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Theorem bj-cbvaldvav 34994
Description: Version of cbvaldva 2411 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-cbvaldvav.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
bj-cbvaldvav (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem bj-cbvaldvav
StepHypRef Expression
1 nfv 1921 . 2 𝑦𝜑
2 nfvd 1922 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 bj-cbvaldvav.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 413 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4bj-cbvaldv 34990 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1787  df-nf 1791
This theorem is referenced by: (None)
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