Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-equsal Structured version   Visualization version   GIF version

Theorem wl-equsal 33456
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 33455 first, and then deriving more specialized versions wl-equsal 33456 and wl-equsal1t 33457 then is more efficient than the other way round, which is possible, too. See also equsal 2327. (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 1770 . . 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 33455 . 2 (⊤ → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓))
76trud 1533 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wtru 1524  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator