Theorem altxpeq2 32081

Description: Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)

Assertion

Ref	Expression
altxpeq2	|- ( A = B -> ( C XX. A ) = ( C XX. B ) )

Proof of Theorem altxpeq2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 3139	. . . 4 |- ( A = B -> ( E. y e. A z = << x , y >> <-> E. y e. B z = << x , y >> ) )
2	1	rexbidv 3052	. . 3 |- ( A = B -> ( E. x e. C E. y e. A z = << x , y >> <-> E. x e. C E. y e. B z = << x , y >> ) )
3	2	abbidv 2741	. 2 |- ( A = B -> { z | E. x e. C E. y e. A z = << x , y >> } = { z | E. x e. C E. y e. B z = << x , y >> } )
4		df-altxp 32066	. 2 |- ( C XX. A ) = { z | E. x e. C E. y e. A z = << x , y >> }
5		df-altxp 32066	. 2 |- ( C XX. B ) = { z | E. x e. C E. y e. B z = << x , y >> }
6	3, 4, 5	3eqtr4g 2681	1 |- ( A = B -> ( C XX. A ) = ( C XX. B ) )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   <<caltop 32063   XX. caltxp 32064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-altxp 32066

This theorem is referenced by: (None)