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Theorem wl-nfimf1 33293
Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1824 in dvelimdf 2334 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
Assertion
Ref Expression
wl-nfimf1 (∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))

Proof of Theorem wl-nfimf1
StepHypRef Expression
1 nfa1 2027 . 2 𝑥𝑥𝜑
2 pm5.5 351 . . 3 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
32sps 2054 . 2 (∀𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
41, 3nfbidf 2091 1 (∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1704  df-nf 1709
This theorem is referenced by: (None)
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