Theorem spimv 2421

Description: A version of spim 2418 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2274 and spimvw 2006 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2403. Use spimvw 2006 instead. (New usage is discouraged.)

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		nfv 1934	. 2 ⊢ Ⅎ𝑥𝜓
2		spimv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spim 2418	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804

This theorem is referenced by:  spv  2424