Theorem elabgf 2948

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 2376	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3	1, 2	nfel 2383	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
4		elabgf.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfbi 1637	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
6		eleq1 2294	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
7		elabgf.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	bibi12d 235	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)))
9		abid 2219	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
10	1, 5, 8, 9	vtoclgf 2862	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  Ⅎwnf 1508   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  elabf  2949  elabg  2952  elab3gf  2956  elrabf  2960  bj-intabssel  16385