Theorem spim 1752

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1752 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)

Hypotheses

Ref	Expression
spim.1	⊢ Ⅎ𝑥𝜓
spim.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spim	⊢ (∀𝑥𝜑 → 𝜓)

Proof of Theorem spim

Step	Hyp	Ref	Expression
1		spim.1	. . 3 ⊢ Ⅎ𝑥𝜓
2	1	nfri 1533	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		spim.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimh 1751	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  cbv3  1756  chvar  1771  spimv  1825  2spim  15496