Theorem cdlemk40f 36207

Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk40.x	|- X = ( iota_ z e. T ph )
cdlemk40.u	|- U = ( g e. T |-> if ( F = N , g , X ) )

Assertion

Ref	Expression
cdlemk40f	|- ( ( F =/= N /\ G e. T ) -> ( U ` G ) = [_ G / g ]_ X )

Distinct variable groups:   g, F   g, N   T, g

Allowed substitution hints:   ph( z, g)   T( z)   U( z, g)   F( z)   G( z, g)   N( z)   X( z, g)

Proof of Theorem cdlemk40f

Step	Hyp	Ref	Expression
1		cdlemk40.x	. . 3 |- X = ( iota_ z e. T ph )
2		cdlemk40.u	. . 3 |- U = ( g e. T |-> if ( F = N , g , X ) )
3	1, 2	cdlemk40 36205	. 2 |- ( G e. T -> ( U ` G ) = if ( F = N , G , [_ G / g ]_ X ) )
4		ifnefalse 4098	. 2 |- ( F =/= N -> if ( F = N , G , [_ G / g ]_ X ) = [_ G / g ]_ X )
5	3, 4	sylan9eqr 2678	1 |- ( ( F =/= N /\ G e. T ) -> ( U ` G ) = [_ G / g ]_ X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   [_csb 3533   ifcif 4086   |-> cmpt 4729   ` cfv 5888   iota_crio 6610

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611

This theorem is referenced by:  cdlemk43N  36251  cdlemk35u  36252  cdlemk55u1  36253  cdlemk39u1  36255  cdlemk19u1  36257