Theorem wl-nfimf1 38209

Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1926 in dvelimdf 2481 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 2186	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2		pm5.5 364	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
3	2	sps 2221	. 2 ⊢ (∀𝑥𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
4	1, 3	nfbidf 2260	1 ⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814

This theorem is used by: (None)