Theorem nfded 37825

Description: A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3697. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2911. (Contributed by NM, 19-Nov-2020.)

Hypotheses

Ref	Expression
nfded.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfded.2	⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)
nfded.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfded	⊢ (𝜑 → Ⅎ𝑥𝐶)

Proof of Theorem nfded

Step	Hyp	Ref	Expression
1		nfded.3	. 2 ⊢ Ⅎ𝑥𝐵
2		nfded.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfnfc1 2906	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
4		nfded.2	. . . 4 ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)
5	3, 4	nfceqdf 2898	. . 3 ⊢ (Ⅎ𝑥𝐴 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶))
6	2, 5	syl 17	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶))
7	1, 6	mpbii 232	1 ⊢ (𝜑 → Ⅎ𝑥𝐶)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  Ⅎwnfc 2883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-cleq 2724  df-nfc 2885

This theorem is referenced by:  nfunidALT  37828