Theorem elabf 3614

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 2903	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf.1	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	2, 3, 4	elabgf 3613	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
6	1, 5	ax-mp 5	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  {cab 2719  Vcvv 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-v 3435

This theorem is referenced by:  dfon2lem1  36022  sdclem2  38122  sdclem1  38123  scottabf  44697