Theorem bj-sblem1

Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	bj-sblem1	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-2	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
2	1	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜒 ) ) )
3		19.23v	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜒 ) ↔ ( ∃ 𝑥 𝜑 → 𝜒 ) )
4	2 3	syl6ib	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜒 ) ) )