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Theorem spimv 2256
Description: A version of spim 2253 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2112 and spimvw 1924. See also spimvALT 2257. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2016. (Revised by BJ, 29-Nov-2020.)
Hypothesis
Ref Expression
spimv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv (∀𝑥𝜑𝜓)
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem spimv
StepHypRef Expression
1 ax6e 2249 . . 3 𝑥 𝑥 = 𝑦
2 spimv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2eximii 1761 . 2 𝑥(𝜑𝜓)
4319.36iv 1902 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  spv  2259  aevALTOLD  2320  axc16i  2321  reu6  3377  el  4807  aev-o  33693  axc11next  38086
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