Theorem abbi2dv 2359

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 1927	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
3		abeq2 2347	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {cab 2224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  sbab  2368  iftrue  3642  iffalse  3645  iniseg  5154  fncnvima2  5821  isoini  6014  dftpos3  6523  unfiexmid  7215  tgval3  15082  txrest  15300  cnblcld  15559