Theorem cbvoprab12 6042

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
cbvoprab12.1	|- F/ w ph
cbvoprab12.2	|- F/ v ph
cbvoprab12.3	|- F/ x ps
cbvoprab12.4	|- F/ y ps
cbvoprab12.5	|- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab12	|- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Distinct variable group:   x, y, z, w, v

Allowed substitution hints:   ph( x, y, z, w, v)   ps( x, y, z, w, v)

Proof of Theorem cbvoprab12

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1552	. . . . 5 |- F/ w u = <. x , y >.
2		cbvoprab12.1	. . . . 5 |- F/ w ph
3	1, 2	nfan 1589	. . . 4 |- F/ w ( u = <. x , y >. /\ ph )
4		nfv 1552	. . . . 5 |- F/ v u = <. x , y >.
5		cbvoprab12.2	. . . . 5 |- F/ v ph
6	4, 5	nfan 1589	. . . 4 |- F/ v ( u = <. x , y >. /\ ph )
7		nfv 1552	. . . . 5 |- F/ x u = <. w , v >.
8		cbvoprab12.3	. . . . 5 |- F/ x ps
9	7, 8	nfan 1589	. . . 4 |- F/ x ( u = <. w , v >. /\ ps )
10		nfv 1552	. . . . 5 |- F/ y u = <. w , v >.
11		cbvoprab12.4	. . . . 5 |- F/ y ps
12	10, 11	nfan 1589	. . . 4 |- F/ y ( u = <. w , v >. /\ ps )
13		opeq12 3835	. . . . . 6 |- ( ( x = w /\ y = v ) -> <. x , y >. = <. w , v >. )
14	13	eqeq2d 2219	. . . . 5 |- ( ( x = w /\ y = v ) -> ( u = <. x , y >. <-> u = <. w , v >. ) )
15		cbvoprab12.5	. . . . 5 |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )
16	14, 15	anbi12d 473	. . . 4 |- ( ( x = w /\ y = v ) -> ( ( u = <. x , y >. /\ ph ) <-> ( u = <. w , v >. /\ ps ) ) )
17	3, 6, 9, 12, 16	cbvex2 1947	. . 3 |- ( E. x E. y ( u = <. x , y >. /\ ph ) <-> E. w E. v ( u = <. w , v >. /\ ps ) )
18	17	opabbii 4127	. 2 |- { <. u , z >. | E. x E. y ( u = <. x , y >. /\ ph ) } = { <. u , z >. | E. w E. v ( u = <. w , v >. /\ ps ) }
19		dfoprab2 6015	. 2 |- { <. <. x , y >. , z >. | ph } = { <. u , z >. | E. x E. y ( u = <. x , y >. /\ ph ) }
20		dfoprab2 6015	. 2 |- { <. <. w , v >. , z >. | ps } = { <. u , z >. | E. w E. v ( u = <. w , v >. /\ ps ) }
21	18, 19, 20	3eqtr4i 2238	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1484   E.wex 1516   <.cop 3646   {copab 4120   {coprab 5968

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-oprab 5971

This theorem is referenced by:  cbvoprab12v  6043  cbvmpox  6046  dfoprab4f  6302  fmpox  6309  tposoprab  6389