Theorem elabgf 2984

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 2492	. . . 4 ⊢ Ⅎx{x ∣ φ}
3	1, 2	nfel 2498	. . 3 ⊢ Ⅎx A ∈ {x ∣ φ}
4		elabgf.2	. . 3 ⊢ Ⅎxψ
5	3, 4	nfbi 1834	. 2 ⊢ Ⅎx(A ∈ {x ∣ φ} ↔ ψ)
6		eleq1 2413	. . 3 ⊢ (x = A → (x ∈ {x ∣ φ} ↔ A ∈ {x ∣ φ}))
7		elabgf.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
8	6, 7	bibi12d 312	. 2 ⊢ (x = A → ((x ∈ {x ∣ φ} ↔ φ) ↔ (A ∈ {x ∣ φ} ↔ ψ)))
9		abid 2341	. 2 ⊢ (x ∈ {x ∣ φ} ↔ φ)
10	1, 5, 8, 9	vtoclgf 2914	1 ⊢ (A ∈ B → (A ∈ {x ∣ φ} ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  elabf  2985  elabg  2987  elab3gf  2991  elrabf  2994