Theorem eqabdv 2469

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Hypothesis

Ref	Expression
eqabdv.1	⊢ (φ → (x ∈ A ↔ ψ))

Assertion

Ref	Expression
eqabdv	⊢ (φ → A = {x ∣ ψ})

Distinct variable groups:   x,A   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem eqabdv

Step	Hyp	Ref	Expression
1		eqabdv.1	. . 3 ⊢ (φ → (x ∈ A ↔ ψ))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(x ∈ A ↔ ψ))
3		eqabb 2459	. 2 ⊢ (A = {x ∣ ψ} ↔ ∀x(x ∈ A ↔ ψ))
4	2, 3	sylibr 203	1 ⊢ (φ → A = {x ∣ ψ})

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by:  sbab  2476  iftrue  3669  iffalse  3670  phialllem1  4617  isoini  5498