Theorem nfded 40024

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 3660. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2931. (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 2926	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
4		nfded.2	. . . 4 ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)
5	3, 4	nfceqdf 2919	. . 3 ⊢ (Ⅎ𝑥𝐴 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶))
6	2, 5	syl 18	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶))
7	1, 6	mpbii 236	1 ⊢ (𝜑 → Ⅎ𝑥𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnfc 2908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-cleq 2753  df-nfc 2910

This theorem is used by:  nfunidALT  40027