Theorem abbi1dv 2286

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 1862	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
3		abeq1 2276	. 2 ⊢ ({𝑥 ∣ 𝜓} = 𝐴 ↔ ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
4	2, 3	sylibr 133	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161

This theorem is referenced by:  abidnf  2894  csbtt  3057  csbvarg  3073  csbie2g  3095  abvor0dc  3432  iinxsng  3939  shftuz  10759