Theorem bj-spimvwt

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

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

Proof

Step	Hyp	Ref	Expression
1		alequexv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ∃ 𝑥 ( 𝜑 → 𝜓 ) )
2		19.36v	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )
3	1 2	sylib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )