Theorem bj-sblem1 34166

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

Assertion

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

Distinct variable group:   𝜒,𝑥

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

Proof of Theorem bj-sblem1

Step	Hyp	Ref	Expression
1		ax-2 7	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2	1	al2imi 1816	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜒)))
3		19.23v 1943	. 2 ⊢ (∀𝑥(𝜑 → 𝜒) ↔ (∃𝑥𝜑 → 𝜒))
4	2, 3	syl6ib 253	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒)))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  bj-sbievw1  34169