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Theorem spimv 2293
Description: A version of spim 2290 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2153 and spimvw 1973. See also spimvALT 2294. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2059. (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 2286 . . 3 𝑥 𝑥 = 𝑦
2 spimv.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  ax-7 1981  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  spv  2296  axc16i  2353  reu6  3428  el  4877  aev-o  34535  axc11next  38924
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