Theorem bj-ax12w

Description: The general statement that ax12w proves. (Contributed by BJ, 20-Mar-2020)

	Ref	Expression
Hypotheses	bj-ax12w.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bj-ax12w.2	⊢ ( 𝑦 = 𝑧 → ( 𝜓 ↔ 𝜃 ) )
Assertion	bj-ax12w	⊢ ( 𝜑 → ( ∀ 𝑦 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12w.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bj-ax12w.2	⊢ ( 𝑦 = 𝑧 → ( 𝜓 ↔ 𝜃 ) )
3	2	spw	⊢ ( ∀ 𝑦 𝜓 → 𝜓 )
4	1	bj-ax12wlem	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
5	3 4	syl5	⊢ ( 𝜑 → ( ∀ 𝑦 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )