Theorem abbi2dv 2312

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 1885	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓))
3		abeq2 2302	. 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 1362   = wceq 1364   ∈ wcel 2164  {cab 2179

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189

This theorem is referenced by:  sbab  2321  iftrue  3562  iffalse  3565  iniseg  5037  fncnvima2  5679  isoini  5861  dftpos3  6315  unfiexmid  6974  tgval3  14226  txrest  14444  cnblcld  14703