Theorem wl-nfimf1 35612

Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1900 in dvelimdf 2449 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)

Assertion

Ref	Expression
wl-nfimf1	⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓))

Proof of Theorem wl-nfimf1

Step	Hyp	Ref	Expression
1		nfa1 2150	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2		pm5.5 361	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
3	2	sps 2180	. 2 ⊢ (∀𝑥𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
4	1, 3	nfbidf 2220	1 ⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)