Theorem bj-ax12ig 37302

Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 37303. (Contributed by BJ, 19-Dec-2020.)

Hypotheses

Ref	Expression
bj-ax12ig.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bj-ax12ig.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))

Assertion

Ref	Expression
bj-ax12ig	⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem bj-ax12ig

Step	Hyp	Ref	Expression
1		bj-ax12ig.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	pm5.32i 585	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
3		bj-ax12ig.2	. . . . 5 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
4	3	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥𝜒)
5	1	biimprcd 253	. . . 4 ⊢ (𝜒 → (𝜑 → 𝜓))
6	4, 5	sylg 1856	. . 3 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓))
7	2, 6	sylbi 220	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓))
8	7	ex 418	1 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  bj-ax12i  37303