Theorem nfcxfrd 2488

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 2486	. 2 ⊢ (ℲxA ↔ ℲxB)
4	1, 3	sylibr 203	1 ⊢ (φ → ℲxA)

Syntax hints:   → wi 4   = wceq 1642  Ⅎwnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfcsb1d  3167  nfcsbd  3170  nfifd  3686  nfunid  3899  nfiotad  4343  nfovd  5545