Theorem bj-ax12ig

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

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

Proof

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