Theorem nfcxfrd 2969

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfceqi.1	⊢ 𝐴 = 𝐵
nfcxfrd.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfcxfrd	⊢ (𝜑 → Ⅎ𝑥𝐴)

Proof of Theorem nfcxfrd

Step	Hyp	Ref	Expression
1		nfcxfrd.2	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
2		nfceqi.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	nfceqi 2967	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 226	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1658  Ⅎwnfc 2957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-nf 1885  df-cleq 2819  df-clel 2822  df-nfc 2959

This theorem is referenced by:  nfcsb1d  3772  nfcsbd  3775  nfifd  4335  nfunid  4666  nfiotad  6090  nfriotad  6875  nfovd  6935  nfnegd  10597  nfxnegd  40464  nfintd  43316  nfiund  43317