Theorem coeq1 2502

Description: Equality theorem for composition of two classes.

Assertion

Ref	Expression
coeq1	|- (A = B -> (A o. C) = (B o. C))

Proof of Theorem coeq1

Step	Hyp	Ref	Expression
1		breq 2064	. . . . 5 |- (A = B -> (zAy <-> zBy))
2	1	anbi2d 468	. . . 4 |- (A = B -> ((xCz /\ zAy) <-> (xCz /\ zBy)))
3	2	biexdv 936	. . 3 |- (A = B -> (E.z(xCz /\ zAy) <-> E.z(xCz /\ zBy)))
4	3	biopabdv 2102	. 2 |- (A = B -> {<.x, y>. | E.z(xCz /\ zAy)} = {<.x, y>. | E.z(xCz /\ zBy)})
5		df-co 2427	. 2 |- (A o. C) = {<.x, y>. | E.z(xCz /\ zAy)}
6		df-co 2427	. 2 |- (B o. C) = {<.x, y>. | E.z(xCz /\ zBy)}
7	4, 5, 6	3eqtr4g 1147	1 |- (A = B -> (A o. C) = (B o. C))

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   class class class wbr 2054  {copab 2055   o. ccom 2414

This theorem is referenced by:  coeq1i 2504  coeq1d 2506  coi2 2666  ereq 3206

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-br 2063  df-opab 2098  df-co 2427

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