Theorem bj-ax12wlem 34727

Description: A lemma used to prove a weak version of the axiom of substitution ax-12 2177. (Temporary comment: The general statement that ax12wlem 2134 proves.) (Contributed by BJ, 20-Mar-2020.)

Hypothesis

Ref	Expression
bj-ax12wlem.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-ax12wlem	⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

Distinct variable group:   𝜒,𝑥

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

Proof of Theorem bj-ax12wlem

Step	Hyp	Ref	Expression
1		bj-ax12wlem.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		ax-5 1918	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
3	1, 2	bj-ax12i 34720	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  ax-5 1918

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

This theorem is referenced by:  bj-ax12w  34760