Theorem unceq 33386

Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
unceq	|- ( A = B -> uncurry A = uncurry B )

Proof of Theorem unceq

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

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . 4 |- ( A = B -> ( A ` x ) = ( B ` x ) )
2	1	breqd 4664	. . 3 |- ( A = B -> ( y ( A ` x ) z <-> y ( B ` x ) z ) )
3	2	oprabbidv 6709	. 2 |- ( A = B -> { <. <. x , y >. , z >. | y ( A ` x ) z } = { <. <. x , y >. , z >. | y ( B ` x ) z } )
4		df-unc 7394	. 2 |- uncurry A = { <. <. x , y >. , z >. | y ( A ` x ) z }
5		df-unc 7394	. 2 |- uncurry B = { <. <. x , y >. , z >. | y ( B ` x ) z }
6	3, 4, 5	3eqtr4g 2681	1 |- ( A = B -> uncurry A = uncurry B )

Syntax hints:   -> wi 4   = wceq 1483   class class class wbr 4653   ` cfv 5888   {coprab 6651  uncurry cunc 7392

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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-oprab 6654  df-unc 7394

This theorem is referenced by: (None)