Theorem wl-equsal 33326

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 33325 first, and then deriving more specialized versions wl-equsal 33326 and wl-equsal1t 33327 then is more efficient than the other way round, which is possible, too. See also equsal 2291. (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 1730	. . 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 33325	. 2 ⊢ (⊤ → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓))
7	6	trud 1493	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ⊤wtru 1484  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)