Theorem spim 2408

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2408 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 2404	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spim.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1937	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2276	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1656  Ⅎwnf 1884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-12 2222  ax-13 2391

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-nf 1885

This theorem is referenced by:  spimv  2411  chvar  2416  cbv3  2418  setrec2fun  43334