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Theorem nfan1c 35041
Description: Variant of nfan 1898 and commuted form of nfan1 2196. (Contributed by BTernaryTau, 31-Jul-2025.)
Hypotheses
Ref Expression
nfan1c.1 𝑥𝜑
nfan1c.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfan1c 𝑥(𝜓𝜑)

Proof of Theorem nfan1c
StepHypRef Expression
1 nfan1c.1 . . 3 𝑥𝜑
2 nfan1c.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
31, 2nfan1 2196 . 2 𝑥(𝜑𝜓)
4 ancom 460 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
54nfbii 1850 . 2 (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥(𝜓𝜑))
63, 5mpbi 230 1 𝑥(𝜓𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782
This theorem is referenced by:  dvelimalcased  35043  dvelimexcased  35045
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