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Theorem wl-equsal1t 33809
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 2401, spimt 2370 or sbft 2477, 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 2198 . 2 𝑥𝑥𝜑
2 id 22 . 2 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 biid 253 . . 3 (𝜑𝜑)
432a1i 12 . 2 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑𝜑)))
51, 2, 4wl-equsald 33807 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213  ax-13 2354
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880
This theorem is referenced by:  wl-equsal1i  33811
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