Theorem spimd 16793

Description: Deduction form of spim 1791. (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 1789	. 2 ⊢ ((Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) → (∀𝑥𝜓 → 𝜒))
4	1, 2, 3	syl2anc 415	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  Ⅎwnf 1513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  2spim  16794