Theorem bnj976 31769

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

Hypotheses

Ref	Expression
bnj976.1	|- ( ch <-> ( N e. D /\ f Fn N /\ ph /\ ps ) )
bnj976.2	|- ( ph' <-> [. G / f ]. ph )
bnj976.3	|- ( ps' <-> [. G / f ]. ps )
bnj976.4	|- ( ch' <-> [. G / f ]. ch )
bnj976.5	|- G e. _V

Assertion

Ref	Expression
bnj976	|- ( ch' <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) )

Distinct variable groups:   D, f   f, N

Allowed substitution hints:   ph( f)   ps( f)   ch( f)   G( f)   ph'( f)   ps'( f)   ch'( f)

Proof of Theorem bnj976

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj976.4	. 2 |- ( ch' <-> [. G / f ]. ch )
2		sbcco 3208	. 2 |- ( [. G / h ]. [. h / f ]. ch <-> [. G / f ]. ch )
3		bnj976.5	. . 3 |- G e. _V
4		bnj252 31689	. . . . . 6 |- ( ( N e. D /\ f Fn N /\ ph /\ ps ) <-> ( N e. D /\ ( f Fn N /\ ph /\ ps ) ) )
5	4	sbcbii 3245	. . . . 5 |- ( [. h / f ]. ( N e. D /\ f Fn N /\ ph /\ ps ) <-> [. h / f ]. ( N e. D /\ ( f Fn N /\ ph /\ ps ) ) )
6		bnj976.1	. . . . . 6 |- ( ch <-> ( N e. D /\ f Fn N /\ ph /\ ps ) )
7	6	sbcbii 3245	. . . . 5 |- ( [. h / f ]. ch <-> [. h / f ]. ( N e. D /\ f Fn N /\ ph /\ ps ) )
8		vex 2974	. . . . . . . 8 |- h e. _V
9	8	bnj525 31728	. . . . . . 7 |- ( [. h / f ]. N e. D <-> N e. D )
10		sbc3an 3248	. . . . . . . 8 |- ( [. h / f ]. ( f Fn N /\ ph /\ ps ) <-> ( [. h / f ]. f Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
11		bnj62 31707	. . . . . . . . 9 |- ( [. h / f ]. f Fn N <-> h Fn N )
12	11	3anbi1i 1178	. . . . . . . 8 |- ( ( [. h / f ]. f Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
13	10, 12	bitri 249	. . . . . . 7 |- ( [. h / f ]. ( f Fn N /\ ph /\ ps ) <-> ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
14	9, 13	anbi12i 697	. . . . . 6 |- ( ( [. h / f ]. N e. D /\ [. h / f ]. ( f Fn N /\ ph /\ ps ) ) <-> ( N e. D /\ ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) ) )
15		sbcan 3228	. . . . . 6 |- ( [. h / f ]. ( N e. D /\ ( f Fn N /\ ph /\ ps ) ) <-> ( [. h / f ]. N e. D /\ [. h / f ]. ( f Fn N /\ ph /\ ps ) ) )
16		bnj252 31689	. . . . . 6 |- ( ( N e. D /\ h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> ( N e. D /\ ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) ) )
17	14, 15, 16	3bitr4ri 278	. . . . 5 |- ( ( N e. D /\ h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> [. h / f ]. ( N e. D /\ ( f Fn N /\ ph /\ ps ) ) )
18	5, 7, 17	3bitr4i 277	. . . 4 |- ( [. h / f ]. ch <-> ( N e. D /\ h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
19		fneq1 5498	. . . . . . 7 |- ( h = G -> ( h Fn N <-> G Fn N ) )
20		sbceq1a 3196	. . . . . . . 8 |- ( h = G -> ( [. h / f ]. ph <-> [. G / h ]. [. h / f ]. ph ) )
21		bnj976.2	. . . . . . . . 9 |- ( ph' <-> [. G / f ]. ph )
22		sbcco 3208	. . . . . . . . 9 |- ( [. G / h ]. [. h / f ]. ph <-> [. G / f ]. ph )
23	21, 22	bitr4i 252	. . . . . . . 8 |- ( ph' <-> [. G / h ]. [. h / f ]. ph )
24	20, 23	syl6bbr 263	. . . . . . 7 |- ( h = G -> ( [. h / f ]. ph <-> ph' ) )
25		sbceq1a 3196	. . . . . . . 8 |- ( h = G -> ( [. h / f ]. ps <-> [. G / h ]. [. h / f ]. ps ) )
26		bnj976.3	. . . . . . . . 9 |- ( ps' <-> [. G / f ]. ps )
27		sbcco 3208	. . . . . . . . 9 |- ( [. G / h ]. [. h / f ]. ps <-> [. G / f ]. ps )
28	26, 27	bitr4i 252	. . . . . . . 8 |- ( ps' <-> [. G / h ]. [. h / f ]. ps )
29	25, 28	syl6bbr 263	. . . . . . 7 |- ( h = G -> ( [. h / f ]. ps <-> ps' ) )
30	19, 24, 29	3anbi123d 1289	. . . . . 6 |- ( h = G -> ( ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> ( G Fn N /\ ph' /\ ps' ) ) )
31	30	anbi2d 703	. . . . 5 |- ( h = G -> ( ( N e. D /\ ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) ) <-> ( N e. D /\ ( G Fn N /\ ph' /\ ps' ) ) ) )
32		bnj252 31689	. . . . 5 |- ( ( N e. D /\ G Fn N /\ ph' /\ ps' ) <-> ( N e. D /\ ( G Fn N /\ ph' /\ ps' ) ) )
33	31, 16, 32	3bitr4g 288	. . . 4 |- ( h = G -> ( ( N e. D /\ h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) ) )
34	18, 33	syl5bb 257	. . 3 |- ( h = G -> ( [. h / f ]. ch <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) ) )
35	3, 34	sbcie 3220	. 2 |- ( [. G / h ]. [. h / f ]. ch <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) )
36	1, 2, 35	3bitr2i 273	1 |- ( ch' <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) )

Syntax hints:   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   _Vcvv 2971   [.wsbc 3185   Fn wfn 5412   /\ w-bnj17 31672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5419  df-fn 5420  df-bnj17 31673

This theorem is referenced by:  bnj910  31939  bnj999  31948  bnj907  31956