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Theorem spimv 1791
 Description: A version of spim 1718 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.)
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 1508 . 2 𝑥𝜓
2 spimv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spim 1718 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1333 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1441 This theorem is referenced by:  aev  1792  ax16i  1838  spv  1840  cbvalvw  1899  reu6  2901  el  4138
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