Theorem elab2 2989

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	⊢ A ∈ V
elab2.2	⊢ (x = A → (φ ↔ ψ))
elab2.3	⊢ B = {x ∣ φ}

Assertion

Ref	Expression
elab2	⊢ (A ∈ B ↔ ψ)

Distinct variable groups:   ψ,x   x,A

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

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 ⊢ A ∈ V
2		elab2.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3		elab2.3	. . 3 ⊢ B = {x ∣ φ}
4	2, 3	elab2g 2988	. 2 ⊢ (A ∈ V → (A ∈ B ↔ ψ))
5	1, 4	ax-mp 5	1 ⊢ (A ∈ B ↔ ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860

This proof depends on 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 proof 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 used by:  elpw  3729  elint  3933  opkelopkabg  4246  0ceven  4506  eventfin  4518  oddtfin  4519  dfphi2  4570  phi11lem1  4596  0cnelphi  4598  proj1op  4601  proj2op  4602  opabid  4696  oprabid  5551  elfuns  5830