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Theorem bj-spimvv 32846
Description: Version of spimv 2293 and spimv1 2153 with a dv condition, which does not require ax-13 2282. UPDATE: this is spimvw 1973. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-spimvv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-spimvv (∀𝑥𝜑𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-spimvv
StepHypRef Expression
1 ax6ev 1947 . . 3 𝑥 𝑥 = 𝑦
2 bj-spimvv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2eximii 1804 . 2 𝑥(𝜑𝜓)
4319.36iv 1914 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  bj-spvv  32848  bj-el  32921
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