Theorem bj-spimt2

Description: A step in the proof of spimt . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-spimt2	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-alequex	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ∃ 𝑥 ( 𝜑 → 𝜓 ) )
2		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	1 2	sylib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
4	3	imim1d	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )