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Theorem wl-equsal 34312
 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 34311 first, and then deriving more specialized versions wl-equsal 34312 and wl-equsal1t 34313 then is more efficient than the other way round, which is possible, too. See also equsal 2395. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
wl-equsal.1 𝑥𝜓
wl-equsal.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
wl-equsal (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem wl-equsal
StepHypRef Expression
1 nftru 1786 . . 3 𝑥
2 wl-equsal.1 . . . 4 𝑥𝜓
32a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜓)
4 wl-equsal.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
54a1i 11 . . 3 (⊤ → (𝑥 = 𝑦 → (𝜑𝜓)))
61, 3, 5wl-equsald 34311 . 2 (⊤ → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓))
76mptru 1529 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520  ⊤wtru 1523  Ⅎwnf 1765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1525  df-ex 1762  df-nf 1766 This theorem is referenced by: (None)
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