Theorem abeq2i 2288

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)

Hypothesis

Ref	Expression
abeqi.1	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
abeq2i	⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Proof of Theorem abeq2i

Step	Hyp	Ref	Expression
1		abeqi.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
2	1	eleq2i 2244	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2165	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173

This theorem is referenced by:  rabid  2653  vex  2742  csbco  3069  csbcow  3070  csbnestgf  3111  ifmdc  3576  pwss  3593  snsspw  3766  iunpw  4482  ordon  4487  funcnv3  5280  tfrlem4  6316  tfrlem8  6321  tfrlem9  6322  tfrlemibxssdm  6330  tfr1onlembxssdm  6346  tfrcllembxssdm  6359  ixpm  6732  mapsnen  6813  sbthlem1  6958  1idprl  7591  1idpru  7592  recexprlem1ssl  7634  recexprlem1ssu  7635  recexprlemss1l  7636  recexprlemss1u  7637  txbas  13843