Theorem spimh 1625

Description: Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1626 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spimh.1	⊢ (𝜓 → ∀𝑥𝜓)
spimh.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimh	⊢ (∀𝑥𝜑 → 𝜓)

Proof of Theorem spimh

Step	Hyp	Ref	Expression
1		spimh.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
2		spimh.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
3	1, 2	syl6com 31	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓))
4	3	alimi 1344	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
5		ax9o 1588	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
6	4, 5	syl 14	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  spim  1626