Theorem opelopaba 4296

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopaba.1	|- A e. _V
opelopaba.2	|- B e. _V
opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
opelopaba	|- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )

Distinct variable groups:   x, y, A   x, B, y   ps, x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem opelopaba

Step	Hyp	Ref	Expression
1		opelopaba.1	. 2 |- A e. _V
2		opelopaba.2	. 2 |- B e. _V
3		opelopaba.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4	3	opelopabga 4293	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
5	1, 2, 4	mp2an 426	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3621   {copab 4089

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091

This theorem is referenced by: (None)