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Theorem spimd 12961
Description: Deduction form of spim 1716. (Contributed by BJ, 17-Oct-2019.)
Hypotheses
Ref Expression
spimd.nf (𝜑 → Ⅎ𝑥𝜒)
spimd.1 (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
spimd (𝜑 → (∀𝑥𝜓𝜒))

Proof of Theorem spimd
StepHypRef Expression
1 spimd.nf . 2 (𝜑 → Ⅎ𝑥𝜒)
2 spimd.1 . 2 (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒)))
3 spimt 1714 . 2 ((Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒))) → (∀𝑥𝜓𝜒))
41, 2, 3syl2anc 408 1 (𝜑 → (∀𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wnf 1436
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  2spim  12962
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