Theorem nfcxfrdf 38664

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)

Hypotheses

Ref	Expression
nfcxfrdf.0	⊢ Ⅎ𝑥𝜑
nfcxfrdf.1	⊢ (𝜑 → 𝐴 = 𝐵)
nfcxfrdf.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfcxfrdf	⊢ (𝜑 → Ⅎ𝑥𝐴)

Proof of Theorem nfcxfrdf

Step	Hyp	Ref	Expression
1		nfcxfrdf.2	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
2		nfcxfrdf.0	. . 3 ⊢ Ⅎ𝑥𝜑
3		nfcxfrdf.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	nfceqdf 2887	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	1, 4	mpbird 256	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1534  Ⅎwnf 1778  Ⅎwnfc 2876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-nf 1779  df-cleq 2718  df-nfc 2878

This theorem is referenced by: (None)