Theorem nfxfrd 1486

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nfbii.1	⊢ (𝜑 ↔ 𝜓)
nfxfrd.2	⊢ (𝜒 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfxfrd	⊢ (𝜒 → Ⅎ𝑥𝜑)

Proof of Theorem nfxfrd

Step	Hyp	Ref	Expression
1		nfxfrd.2	. 2 ⊢ (𝜒 → Ⅎ𝑥𝜓)
2		nfbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	nfbii 1484	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
4	1, 3	sylibr 134	1 ⊢ (𝜒 → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  nf3and  1580  nfbid  1599  nfsbxy  1954  nfsbxyt  1955  nfeud  2054  nfmod  2055  nfeqd  2347  nfeld  2348  nfabdw  2351  nfabd  2352  nfned  2454  nfneld  2463  nfraldw  2522  nfraldxy  2523  nfrexdxy  2524  nfraldya  2525  nfrexdya  2526  nfsbc1d  2994  nfsbcd  2997  nfsbcdw  3106  nfbrd  4063