Theorem elabgf 3598

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
elabgf.1	⊢ Ⅎ𝑥𝐴
elabgf.2	⊢ Ⅎ𝑥𝜓
elabgf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elabgf	⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Proof of Theorem elabgf

Step	Hyp	Ref	Expression
1		elabgf.1	. 2 ⊢ Ⅎ𝑥𝐴
2		nfab1 2908	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3	1, 2	nfel 2920	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
4		elabgf.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfbi 1907	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
6		eleq1 2826	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
7		elabgf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	bibi12d 345	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)))
9		abid 2719	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
10	1, 5, 8, 9	vtoclgf 3493	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  {cab 2715  Ⅎwnfc 2886

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:  elabf  3599  elab3gf  3608  elrabf  3613  currysetlem  35061  currysetlem1  35063