Theorem bj-ax12ig 31802

Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 31803. (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 667	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
3		bj-ax12ig.2	. . . . 5 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
4	3	imp 444	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥𝜒)
5	1	biimprcd 239	. . . 4 ⊢ (𝜒 → (𝜑 → 𝜓))
6	4, 5	sylg 1740	. . 3 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓))
7	2, 6	sylbi 206	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓))
8	7	ex 449	1 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by:  bj-ax12i  31803