Theorem spimd 12972

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 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
spimd	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Proof of Theorem spimd

Step	Hyp	Ref	Expression
1		spimd.nf	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
2		spimd.1	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))
3		spimt 1714	. 2 ⊢ ((Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) → (∀𝑥𝜓 → 𝜒))
4	1, 2, 3	syl2anc 408	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1329  Ⅎwnf 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  12973