Theorem bj-sblem2 34954

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 1946	. 2 ⊢ (∀𝑥(𝜑 → 𝜒) ↔ (∃𝑥𝜑 → 𝜒))
2		ax-2 7	. . 3 ⊢ ((𝜑 → (𝜒 → 𝜓)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓)))
3	2	al2imi 1819	. 2 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → (∀𝑥(𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))
4	1, 3	syl5bir 242	1 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  bj-sbievw2  34957