Theorem wl-equsal 32505

Description: A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 32504 first, and then deriving more specialized versions wl-equsal 32505 and wl-equsal1t 32506 then is more efficient than the other way round, which is possible, too. See also equsal 2279. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem wl-equsal

Step	Hyp	Ref	Expression
1		nftru 1721	. . 3 ⊢ Ⅎ𝑥⊤
2		wl-equsal.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜓)
4		wl-equsal.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	a1i 11	. . 3 ⊢ (⊤ → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)))
6	1, 3, 5	wl-equsald 32504	. 2 ⊢ (⊤ → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓))
7	6	trud 1484	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ⊤wtru 1476  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)