Theorem elabf 3599

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
elabf.1	⊢ Ⅎ𝑥𝜓
elabf.2	⊢ 𝐴 ∈ V
elabf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elabf	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabf

Step	Hyp	Ref	Expression
1		elabf.2	. 2 ⊢ 𝐴 ∈ V
2		nfcv 2906	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf.1	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	2, 3, 4	elabgf 3598	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
6	1, 5	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  {cab 2715  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424

This theorem is referenced by:  elabOLD  3603  dfon2lem1  33665  sdclem2  35827  sdclem1  35828  scottabf  41747