Theorem spimh 1725

Description: Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1726 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 35	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓))
4	3	alimi 1443	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
5		ax9o 1686	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
6	4, 5	syl 14	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1341

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  spim  1726