Theorem spim 2242

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2242 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-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	. 2 ⊢ Ⅎ𝑥𝜓
2		ax6e 2238	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spim.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1754	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2086	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  spimvALT  2246  chvar  2250  cbv3  2253  setrec2fun  42238