Theorem ceqsal1t 3513

Description: One direction of ceqsalt 3514 is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft 3551. Extracted from a proof of ceqsalt 3514. (Contributed by Wolf Lammen, 25-Mar-2025.)

Assertion

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

Proof of Theorem ceqsal1t

Step	Hyp	Ref	Expression
1		biimpr 220	. . . . . 6 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	imim2i 16	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
3	2	com23 86	. . . 4 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑)))
4	3	alimi 1810	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
5		19.21t 2205	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
6	4, 5	imbitrid 244	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
7	6	imp 406	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537   = wceq 1539  Ⅎwnf 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783

This theorem is referenced by:  ceqsalt  3514