Theorem nfcxfrdf 39459

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 2898	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	1, 4	mpbird 258	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1547  Ⅎwnf 1790  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-cleq 2732  df-nfc 2889

This theorem is referenced by: (None)