Theorem spimv 2424

Description: A version of spim 2421 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2278 and spimvw 2019 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2406. Use spimvw 2019 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 1947	. 2 ⊢ Ⅎ𝑥𝜓
2		spimv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spim 2421	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  spv  2427