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Theorem spimv 2245
Description: A version of spim 2242 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2101 and spimvw 1914. See also spimvALT 2246. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2006. (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 2238 . . 3 𝑥 𝑥 = 𝑦
2 spimv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2eximii 1754 . 2 𝑥(𝜑𝜓)
4319.36iv 1892 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by:  spv  2248  aevALTOLD  2309  axc16i  2310  reu6  3362  el  4773  aev-o  33234  axc11next  37629
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