Theorem abbi2dv 2218

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 1813	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
3		abeq2 2208	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
4	2, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1297   = wceq 1299   ∈ wcel 1448  {cab 2086

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096

This theorem is referenced by:  sbab  2226  iftrue  3426  iffalse  3429  iniseg  4847  fncnvima2  5473  isoini  5651  dftpos3  6089  unfiexmid  6735  tgval3  12009  txrest  12226  cnblcld  12457