Theorem bj-equsal 37176

Description: Shorter proof of equsal 2421. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2421, but "min */exc equsal" is ok. (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem bj-equsal

Step	Hyp	Ref	Expression
1		bj-equsal.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		bj-equsal.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 230	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	bj-equsal1 37174	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓)
5	2	biimprd 249	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
6	1, 5	bj-equsal2 37175	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7	4, 6	impbii 210	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1541  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-12 2185  ax-13 2376

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)