Theorem nfcxfrd 2976

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

Hypotheses

Ref	Expression
nfcxfr.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		nfcxfr.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	nfceqi 2973	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 236	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by:  nfcsb1d  3905  nfcsbd  3908  nfcsbw  3909  nfifd  4495  nfunid  4844  nfiotadw  6317  nfiotad  6319  nfriotadw  7122  nfriotad  7125  nfovd  7185  nfnegd  10881  nfxnegd  41735  nfintd  44796  nfiund  44797  nfiundg  44798