Theorem bj-sblem2 37677

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

Assertion

Ref	Expression
bj-sblem2	⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))

Distinct variable group:   𝜒,𝑥

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

Proof of Theorem bj-sblem2

Step	Hyp	Ref	Expression
1		19.23v 1975	. 2 ⊢ (∀𝑥(𝜑 → 𝜒) ↔ (∃𝑥𝜑 → 𝜒))
2		ax-2 7	. . 3 ⊢ ((𝜑 → (𝜒 → 𝜓)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓)))
3	2	al2imi 1848	. 2 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → (∀𝑥(𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))
4	1, 3	biimtrrid 246	1 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

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

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-sbievw2  37680