Theorem spim 2405

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2405 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out spimw 1973 for a version requiring less axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.)

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 2401	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spim.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1837	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2233	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by:  spimv  2408  chvar  2413  cbv3  2415