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Theorem speivw 1982
Description: Version of spei 2395 with a disjoint variable condition, which does not require ax-13 2373 (neither ax-7 2016 nor ax-12 2177). (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
speivw.1 (𝑥 = 𝑦 → (𝜑𝜓))
speivw.2 𝜓
Assertion
Ref Expression
speivw 𝑥𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem speivw
StepHypRef Expression
1 speivw.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21biimprd 251 . 2 (𝑥 = 𝑦 → (𝜓𝜑))
3 speivw.2 . 2 𝜓
42, 3speiv 1981 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-6 1976
This theorem depends on definitions:  df-bi 210  df-ex 1788
This theorem is referenced by:  elirrv  9239  bnj1014  32682  eusnsn  44225
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