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Theorem bj-cbvexvv 33040
Description: Version of cbvexv 2420 with a dv condition, which does not require ax-13 2391. UPDATE: this is cbvexvw 2121 (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-cbvexvv (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bj-cbvexvv
StepHypRef Expression
1 nfv 1992 . 2 𝑦𝜑
2 nfv 1992 . 2 𝑥𝜓
3 bj-cbvalvv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvexv1 2321 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859
This theorem is referenced by: (None)
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