Theorem spim 1717

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1717 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 1500	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		spim.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimh 1716	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1330  Ⅎwnf 1437

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438

This theorem is referenced by:  cbv3  1721  chvar  1731  spimv  1784  2spim  13144