Theorem bnj581 30978

Description: Technical lemma for bnj580 30983. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj581.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
bnj581.4	|- ( ph' <-> [. g / f ]. ph )
bnj581.5	|- ( ps' <-> [. g / f ]. ps )
bnj581.6	|- ( ch' <-> [. g / f ]. ch )

Assertion

Ref	Expression
bnj581	|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )

Distinct variable group:   f, n

Allowed substitution hints:   ph( f, g, n)   ps( f, g, n)   ch( f, g, n)   ph'( f, g, n)   ps'( f, g, n)   ch'( f, g, n)

Proof of Theorem bnj581

Step	Hyp	Ref	Expression
1		bnj581.6	. 2 |- ( ch' <-> [. g / f ]. ch )
2		bnj581.3	. . 3 |- ( ch <-> ( f Fn n /\ ph /\ ps ) )
3	2	sbcbii 3491	. 2 |- ( [. g / f ]. ch <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
4		sbc3an 3494	. . 3 |- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
5		bnj62 30786	. . . . 5 |- ( [. g / f ]. f Fn n <-> g Fn n )
6	5	bicomi 214	. . . 4 |- ( g Fn n <-> [. g / f ]. f Fn n )
7		bnj581.4	. . . 4 |- ( ph' <-> [. g / f ]. ph )
8		bnj581.5	. . . 4 |- ( ps' <-> [. g / f ]. ps )
9	6, 7, 8	3anbi123i 1251	. . 3 |- ( ( g Fn n /\ ph' /\ ps' ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
10	4, 9	bitr4i 267	. 2 |- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) )
11	1, 3, 10	3bitri 286	1 |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   [.wsbc 3435   Fn wfn 5883

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-sbc 3436  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-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  bnj580  30983  bnj849  30995