Theorem nfded2 33273

Description: A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g. ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4357) that starts from abidnf 3342. The last is assigned to the inference form (e.g. Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4356) whose hypotheses are satisfied using nfaba1 2756. (Contributed by NM, 19-Nov-2020.)

Hypotheses

Ref	Expression
nfded2.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfded2.2	⊢ (𝜑 → Ⅎ𝑥𝐵)
nfded2.3	⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)
nfded2.4	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
nfded2	⊢ (𝜑 → Ⅎ𝑥𝐷)

Proof of Theorem nfded2

Step	Hyp	Ref	Expression
1		nfded2.4	. 2 ⊢ Ⅎ𝑥𝐶
2		nfded2.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfded2.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4		nfnfc1 2754	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
5		nfnfc1 2754	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
6	4, 5	nfan 1816	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
7		nfded2.3	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)
8	6, 7	nfceqdf 2747	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷))
9	2, 3, 8	syl2anc 691	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷))
10	1, 9	mpbii 222	1 ⊢ (𝜑 → Ⅎ𝑥𝐷)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  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-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740

This theorem is referenced by:  nfopdALT  33276