Theorem bj-equsal1 37487

Description: One direction of equsal 2448. (Contributed by BJ, 30-Sep-2018.)

Hypotheses

Ref	Expression
bj-equsal1.1	⊢ Ⅎ𝑥𝜓
bj-equsal1.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-equsal1	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)

Proof of Theorem bj-equsal1

Step	Hyp	Ref	Expression
1		bj-equsal1.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
2	1	a2i 15	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓))
3	2	alimi 1840	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓))
4		bj-equsal1.1	. . 3 ⊢ Ⅎ𝑥𝜓
5	4	bj-equsal1ti 37486	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓)
6	3, 5	sylib 221	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813

This theorem is used by:  bj-equsal  37489