Theorem abbi1dv 2207

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

Hypothesis

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

Assertion

Ref	Expression
abbi1dv	⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abbi1dv

Step	Hyp	Ref	Expression
1		abbildv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))
2	1	alrimiv 1802	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
3		abeq1 2197	. 2 ⊢ ({𝑥 ∣ 𝜓} = 𝐴 ↔ ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
4	2, 3	sylibr 132	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1287   = wceq 1289   ∈ wcel 1438  {cab 2074

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084

This theorem is referenced by:  abidnf  2781  csbtt  2941  csbvarg  2956  csbie2g  2976  abvor0dc  3304  iinxsng  3798  shftuz  10216