Theorem mpoeq3dv 6011

Description: An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)

Hypothesis

Ref	Expression
mpoeq3dv.1	|- ( ph -> C = D )

Assertion

Ref	Expression
mpoeq3dv	|- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   D( x, y)

Proof of Theorem mpoeq3dv

Step	Hyp	Ref	Expression
1		mpoeq3dv.1	. . 3 |- ( ph -> C = D )
2	1	3ad2ant1 1021	. 2 |- ( ( ph /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 6009	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   e. cmpo 5946

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-oprab 5948  df-mpo 5949

This theorem is referenced by:  ofeqd  6160  prdsex  13101