Theorem ceqsal1t

Description: One direction of ceqsalt is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft . Extracted from a proof of ceqsalt . (Contributed by Wolf Lammen, 25-Mar-2025)

	Ref	Expression
Assertion	ceqsal1t	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
2	1	imim2i	⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) )
3	2	com23	⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
4	3	alimi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
5		19.21t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) )
6	4 5	imbitrid	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) )
7	6	imp	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )