Theorem bj-evaleq 33024

Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq 15862 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-evaleq	|- ( A = B -> Slot A = Slot B )

Proof of Theorem bj-evaleq

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( A = B -> ( f ` A ) = ( f ` B ) )
2	1	mpteq2dv 4745	. 2 |- ( A = B -> ( f e. _V |-> ( f ` A ) ) = ( f e. _V |-> ( f ` B ) ) )
3		df-slot 15861	. 2 |- Slot A = ( f e. _V |-> ( f ` A ) )
4		df-slot 15861	. 2 |- Slot B = ( f e. _V |-> ( f ` B ) )
5	2, 3, 4	3eqtr4g 2681	1 |- ( A = B -> Slot A = Slot B )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  Slot cslot 15856

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-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-slot 15861

This theorem is referenced by:  bj-ndxid  33030