Theorem nfcxfrd 2926

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 2924	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 237	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnfc 2912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-cleq 2757  df-clel 2840  df-nfc 2914

This theorem is used by:  nfcsb1d  3876  nfcsbd  3879  nfcsbw  3880  nfifd  4519  nfunid  4880  nfopabd  5181  nfiotadw  6499  nfiotad  6501  nfriotadw  7384  nfriotad  7387  nfovd  7448  nfttrcld  9686  nfnegd  11469  nfchnd  18691  nfxnegd  46215  nfintd  50510  nfiund  50511  nfiundg  50512