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Theorem spimd 11008
Description: Deduction form of spim 1668. (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 1666 . 2 ((Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒))) → (∀𝑥𝜓𝜒))
41, 2, 3syl2anc 403 1 (𝜑 → (∀𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  2spim  11009
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