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Theorem wl-nfimf1 36902
Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1891 in dvelimdf 2442 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 2140 . 2 𝑥𝑥𝜑
2 pm5.5 361 . . 3 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
32sps 2170 . 2 (∀𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
41, 3nfbidf 2209 1 (∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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