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Theorem spimv 2393
Description: A version of spim 2390 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2237 and spimvw 1993 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2375. Use spimvw 1993 instead. (New usage is discouraged.)
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 nfv 1912 . 2 𝑥𝜓
2 spimv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spim 2390 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781
This theorem is referenced by:  spv  2396
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