Theorem biabrdv 1184

Description: Deduction from a wff to a class abstraction.

Hypothesis

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

Assertion

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

Distinct variable group(s):   x,A   φ,x

Proof of Theorem biabrdv

Step	Hyp	Ref	Expression
1		biabrdv.1	. . 3 ⊢ (φ → (x ∈ A ↔ ψ))
2	1	19.21aiv 943	. 2 ⊢ (φ → ∀x(x ∈ A ↔ ψ))
3		cleqab 1174	. 2 ⊢ (A = {x∣ψ} ↔ ∀x(x ∈ A ↔ ψ))
4	2, 3	sylibr 175	1 ⊢ (φ → A = {x∣ψ})

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  sbab 1188  birabrdv 1648  iftrue 1780  iffalse 1781  iniseg 2619  isoini 2938  pw2en 3348  r1val2 3522  aceq3 3556

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

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