Theorem nfxfrd 1489

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 1487	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
4	1, 3	sylibr 134	1 ⊢ (𝜒 → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1474

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 1461  ax-gen 1463

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nf3and  1583  nfbid  1602  nfsbxy  1961  nfsbxyt  1962  nfeud  2061  nfmod  2062  nfeqd  2354  nfeld  2355  nfabdw  2358  nfabd  2359  nfned  2461  nfneld  2470  nfraldw  2529  nfraldxy  2530  nfrexdxy  2531  nfraldya  2532  nfrexdya  2533  nfsbc1d  3006  nfsbcd  3009  nfsbcdw  3118  nfbrd  4078