Theorem nfcxfrdf 33271

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 2747	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	1, 4	mpbird 246	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1475  Ⅎwnf 1699  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740

This theorem is referenced by: (None)