Theorem spimv 2293

Description: A version of spim 2290 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2153 and spimvw 1973. See also spimvALT 2294. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2059. (Revised by BJ, 29-Nov-2020.)

Hypothesis

Ref	Expression
spimv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem spimv

Step	Hyp	Ref	Expression
1		ax6e 2286	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2		spimv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	eximii 1804	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
4	3	19.36iv 1914	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745

This theorem is referenced by:  spv  2296  axc16i  2353  reu6  3428  el  4877  aev-o  34535  axc11next  38924