Theorem nfcxfrdf 39768

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 2920	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
5	1, 4	mpbird 260	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnf 1812  Ⅎwnfc 2909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-cleq 2754  df-nfc 2911

This theorem is used by: (None)