Theorem bj-ax12i 34720

Description: A weakening of bj-ax12ig 34719 that is sufficient to prove a weak form of the axiom of substitution ax-12 2177. The general statement of which ax12i 1975 is an instance. (Contributed by BJ, 29-Sep-2019.)

Hypotheses

Ref	Expression
bj-ax12i.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bj-ax12i.2	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
bj-ax12i	⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem bj-ax12i

Step	Hyp	Ref	Expression
1		bj-ax12i.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		bj-ax12i.2	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
3	2	a1i 11	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
4	1, 3	bj-ax12ig 34719	1 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  bj-ax12wlem  34727