Theorem nfded2 38969

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 4890) that starts from abidnf 3708. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4889) whose hypotheses are satisfied using nfaba1 2913. (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 2908	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
5		nfnfc1 2908	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
6	4, 5	nfan 1899	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
7		nfded2.3	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)
8	6, 7	nfceqdf 2901	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷))
9	2, 3, 8	syl2anc 584	. 2 ⊢ (𝜑 → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷))
10	1, 9	mpbii 233	1 ⊢ (𝜑 → Ⅎ𝑥𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnfc 2890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2729  df-nfc 2892

This theorem is referenced by:  nfopdALT  38972