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Theorem wl-equsal1t 34962
 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 2428, spimt 2393 or sbft 2267, 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 2155 . 2 𝑥𝑥𝜑
2 id 22 . 2 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 biid 264 . . 3 (𝜑𝜑)
432a1i 12 . 2 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑𝜑)))
51, 2, 4wl-equsald 34960 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  wl-equsal1i  34964
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