Theorem bj-spimvwt 34850

Description: Closed form of spimvw 1999. See also spimt 2386. (Contributed by BJ, 8-Nov-2021.)

Assertion

Ref	Expression
bj-spimvwt	⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

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

Proof of Theorem bj-spimvwt

Step	Hyp	Ref	Expression
1		alequexv 2004	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓))
2		19.36v 1991	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
3	1, 2	sylib 217	1 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)