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Metamath Proof Explorer

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 ) ) )