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Theorem spimv1 2113
Description: Version of spim 2252 with a dv condition, which does not require ax-13 2244. See spimvw 1925 for a version with two dv conditions, requiring fewer axioms, and spimv 2255 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
spimv1.nf 𝑥𝜓
spimv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv1 (∀𝑥𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimv1
StepHypRef Expression
1 spimv1.nf . 2 𝑥𝜓
2 ax6ev 1888 . . 3 𝑥 𝑥 = 𝑦
3 spimv1.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1762 . 2 𝑥(𝜑𝜓)
51, 419.36i 2097 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708
This theorem is referenced by:  cbv3v  2170  bj-chvarv  32700  bj-cbv3v2  32702  wl-cbv3vv  33278
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