Theorem mpteq12i 3845

Description: An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12i.1	|- A = C
mpteq12i.2	|- B = D

Assertion

Ref	Expression
mpteq12i	|- ( x e. A |-> B ) = ( x e. C |-> D )

Proof of Theorem mpteq12i

Step	Hyp	Ref	Expression
1		mpteq12i.1	. . . 4 |- A = C
2	1	a1i 9	. . 3 |- ( T. -> A = C )
3		mpteq12i.2	. . . 4 |- B = D
4	3	a1i 9	. . 3 |- ( T. -> B = D )
5	2, 4	mpteq12dv 3839	. 2 |- ( T. -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	5	trud 1252	1 |- ( x e. A |-> B ) = ( x e. C |-> D )

Syntax hints:   = wceq 1243   T. wtru 1244   |-> cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820

This theorem is referenced by:  offres  5762