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Theorem bj-cbvexdvav 34114
Description: Version of cbvexdva 2424 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-cbvaldvav.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
bj-cbvexdvav (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem bj-cbvexdvav
StepHypRef Expression
1 nfv 1908 . 2 𝑦𝜑
2 nfvd 1909 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 bj-cbvaldvav.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 415 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4bj-cbvexdv 34110 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1773
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-10 2138  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: (None)
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