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Theorem abbi2dv 2468

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

**Hypothesis**
Ref	Expression
abbirdv.1	⊢ (φ → (x ∈ A ↔ ψ))

**Assertion**
Ref	Expression
abbi2dv	⊢ (φ → A = {x ∣ ψ})

Distinct variable groups: x,A φ,x Allowed substitution hint: ψ(x)

**Proof of Theorem abbi2dv**
Step	Hyp	Ref	Expression
1		abbirdv.1	. . 3 ⊢ (φ → (x ∈ A ↔ ψ))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(x ∈ A ↔ ψ))
3		abeq2 2458	. 2 ⊢ (A = {x ∣ ψ} ↔ ∀x(x ∈ A ↔ ψ))
4	2, 3	sylibr 203	1 ⊢ (φ → A = {x ∣ ψ})

Colors of variables: wff setvar class

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

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 2475 iftrue 3668 iffalse 3669 phialllem1 4616 isoini 5497