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Theorem wl-nfimf1 33622
Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1987 in dvelimdf 2497 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 2195 . 2 𝑥𝑥𝜑
2 pm5.5 352 . . 3 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
32sps 2220 . 2 (∀𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
41, 3nfbidf 2260 1 (∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-10 2185  ax-12 2214
This theorem depends on definitions:  df-bi 198  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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