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Theorem wl-equsal1t 32506
 Description: The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑. This theorem is more fundamental than equsal 2279, spimt 2241 or sbft 2367, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)
Assertion
Ref Expression
wl-equsal1t (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Proof of Theorem wl-equsal1t
StepHypRef Expression
1 nfnf1 2018 . 2 𝑥𝑥𝜑
2 id 22 . 2 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 biid 250 . . 3 (𝜑𝜑)
432a1i 12 . 2 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑𝜑)))
51, 2, 4wl-equsald 32504 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  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-10 2006  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:  wl-equsal1i  32508
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