Theorem abbi1dv 2316

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 1888	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
3		abeq1 2306	. 2 ⊢ ({𝑥 ∣ 𝜓} = 𝐴 ↔ ∀𝑥(𝜓 ↔ 𝑥 ∈ 𝐴))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2167  {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by:  abidnf  2932  csbtt  3096  csbvarg  3112  csbie2g  3135  abvor0dc  3474  iinxsng  3990  shftuz  10982