Theorem predeq123 5681

Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
predeq123	|- ( ( R = S /\ A = B /\ X = Y ) -> Pred ( R , A , X ) = Pred ( S , B , Y ) )

Proof of Theorem predeq123

Step	Hyp	Ref	Expression
1		simp2 1062	. . 3 |- ( ( R = S /\ A = B /\ X = Y ) -> A = B )
2		cnveq 5296	. . . . 5 |- ( R = S -> `' R = `' S )
3	2	3ad2ant1 1082	. . . 4 |- ( ( R = S /\ A = B /\ X = Y ) -> `' R = `' S )
4		sneq 4187	. . . . 5 |- ( X = Y -> { X } = { Y } )
5	4	3ad2ant3 1084	. . . 4 |- ( ( R = S /\ A = B /\ X = Y ) -> { X } = { Y } )
6	3, 5	imaeq12d 5467	. . 3 |- ( ( R = S /\ A = B /\ X = Y ) -> ( `' R " { X } ) = ( `' S " { Y } ) )
7	1, 6	ineq12d 3815	. 2 |- ( ( R = S /\ A = B /\ X = Y ) -> ( A i^i ( `' R " { X } ) ) = ( B i^i ( `' S " { Y } ) ) )
8		df-pred 5680	. 2 |- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
9		df-pred 5680	. 2 |- Pred ( S , B , Y ) = ( B i^i ( `' S " { Y } ) )
10	7, 8, 9	3eqtr4g 2681	1 |- ( ( R = S /\ A = B /\ X = Y ) -> Pred ( R , A , X ) = Pred ( S , B , Y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   i^i cin 3573   {csn 4177   `'ccnv 5113   "cima 5117   Predcpred 5679

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-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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  predeq1  5682  predeq2  5683  predeq3  5684  wsuceq123  31760  wlimeq12  31765