Theorem spim 2395

Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2395 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 2380. See spimw 1970 for a version requiring fewer 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 2391	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spim.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1835	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2232	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782

This theorem is referenced by:  spimv  2398  chvar  2403  cbv3  2405