Theorem bnj156 30796

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj156.1	|- ( ze0 <-> ( f Fn 1o /\ ph' /\ ps' ) )
bnj156.2	|- ( ze1 <-> [. g / f ]. ze0 )
bnj156.3	|- ( ph1 <-> [. g / f ]. ph' )
bnj156.4	|- ( ps1 <-> [. g / f ]. ps' )

Assertion

Ref	Expression
bnj156	|- ( ze1 <-> ( g Fn 1o /\ ph1 /\ ps1 ) )

Proof of Theorem bnj156

Step	Hyp	Ref	Expression
1		bnj156.2	. 2 |- ( ze1 <-> [. g / f ]. ze0 )
2		bnj156.1	. . . 4 |- ( ze0 <-> ( f Fn 1o /\ ph' /\ ps' ) )
3	2	sbcbii 3491	. . 3 |- ( [. g / f ]. ze0 <-> [. g / f ]. ( f Fn 1o /\ ph' /\ ps' ) )
4		sbc3an 3494	. . . 4 |- ( [. g / f ]. ( f Fn 1o /\ ph' /\ ps' ) <-> ( [. g / f ]. f Fn 1o /\ [. g / f ]. ph' /\ [. g / f ]. ps' ) )
5		bnj62 30786	. . . . 5 |- ( [. g / f ]. f Fn 1o <-> g Fn 1o )
6		bnj156.3	. . . . . 6 |- ( ph1 <-> [. g / f ]. ph' )
7	6	bicomi 214	. . . . 5 |- ( [. g / f ]. ph' <-> ph1 )
8		bnj156.4	. . . . . 6 |- ( ps1 <-> [. g / f ]. ps' )
9	8	bicomi 214	. . . . 5 |- ( [. g / f ]. ps' <-> ps1 )
10	5, 7, 9	3anbi123i 1251	. . . 4 |- ( ( [. g / f ]. f Fn 1o /\ [. g / f ]. ph' /\ [. g / f ]. ps' ) <-> ( g Fn 1o /\ ph1 /\ ps1 ) )
11	4, 10	bitri 264	. . 3 |- ( [. g / f ]. ( f Fn 1o /\ ph' /\ ps' ) <-> ( g Fn 1o /\ ph1 /\ ps1 ) )
12	3, 11	bitri 264	. 2 |- ( [. g / f ]. ze0 <-> ( g Fn 1o /\ ph1 /\ ps1 ) )
13	1, 12	bitri 264	1 |- ( ze1 <-> ( g Fn 1o /\ ph1 /\ ps1 ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   [.wsbc 3435   Fn wfn 5883   1oc1o 7553

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:  bnj153  30950