Theorem nfrab 3456

Description: A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker nfrabw 3455 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrab.1	⊢ Ⅎ𝑥𝜑
nfrab.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrab	⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

Proof of Theorem nfrab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3420	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2		nftru 1837	. . . 4 ⊢ Ⅎ𝑦⊤
3		nfrab.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4	3	nfcri 2920	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
5		eleq1w 2849	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	4, 5	dvelimnf 2488	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfrab.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
8	7	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
9	6, 8	nfand 1930	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
10	9	adantl 487	. . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
11	2, 10	nfabd2 2951	. . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
12	11	mptru 1577	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	1, 12	nfcxfr 2926	1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401  ∀wal 1568  ⊤wtru 1571  Ⅎwnf 1816   ∈ wcel 2146  {cab 2744  Ⅎwnfc 2913  {crab 3419

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-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420

This theorem is used by: (None)