Theorem abbi2dv 2325

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 1898	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
3		abeq2 2315	. 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 1371   = wceq 1373   ∈ wcel 2177  {cab 2192

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202

This theorem is referenced by:  sbab  2334  iftrue  3580  iffalse  3583  iniseg  5063  fncnvima2  5714  isoini  5900  dftpos3  6361  unfiexmid  7030  tgval3  14605  txrest  14823  cnblcld  15082