Theorem bnj976 29601

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 3292	. 2 |- ( [. G / h ]. [. h / f ]. ch <-> [. G / f ]. ch )
3		bnj976.5	. . 3 |- G e. _V
4		bnj252 29520	. . . . . 6 |- ( ( N e. D /\ f Fn N /\ ph /\ ps ) <-> ( N e. D /\ ( f Fn N /\ ph /\ ps ) ) )
5	4	sbcbii 3325	. . . . 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 3325	. . . . 5 |- ( [. h / f ]. ch <-> [. h / f ]. ( N e. D /\ f Fn N /\ ph /\ ps ) )
8		vex 3050	. . . . . . . 8 |- h e. _V
9	8	bnj525 29559	. . . . . . 7 |- ( [. h / f ]. N e. D <-> N e. D )
10		sbc3an 3327	. . . . . . . 8 |- ( [. h / f ]. ( f Fn N /\ ph /\ ps ) <-> ( [. h / f ]. f Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
11		bnj62 29538	. . . . . . . . 9 |- ( [. h / f ]. f Fn N <-> h Fn N )
12	11	3anbi1i 1200	. . . . . . . 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 253	. . . . . . 7 |- ( [. h / f ]. ( f Fn N /\ ph /\ ps ) <-> ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
14	9, 13	anbi12i 704	. . . . . 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 3312	. . . . . 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 29520	. . . . . 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 282	. . . . 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 281	. . . 4 |- ( [. h / f ]. ch <-> ( N e. D /\ h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) )
19		fneq1 5669	. . . . . . 7 |- ( h = G -> ( h Fn N <-> G Fn N ) )
20		sbceq1a 3280	. . . . . . . 8 |- ( h = G -> ( [. h / f ]. ph <-> [. G / h ]. [. h / f ]. ph ) )
21		bnj976.2	. . . . . . . . 9 |- ( ph' <-> [. G / f ]. ph )
22		sbcco 3292	. . . . . . . . 9 |- ( [. G / h ]. [. h / f ]. ph <-> [. G / f ]. ph )
23	21, 22	bitr4i 256	. . . . . . . 8 |- ( ph' <-> [. G / h ]. [. h / f ]. ph )
24	20, 23	syl6bbr 267	. . . . . . 7 |- ( h = G -> ( [. h / f ]. ph <-> ph' ) )
25		sbceq1a 3280	. . . . . . . 8 |- ( h = G -> ( [. h / f ]. ps <-> [. G / h ]. [. h / f ]. ps ) )
26		bnj976.3	. . . . . . . . 9 |- ( ps' <-> [. G / f ]. ps )
27		sbcco 3292	. . . . . . . . 9 |- ( [. G / h ]. [. h / f ]. ps <-> [. G / f ]. ps )
28	26, 27	bitr4i 256	. . . . . . . 8 |- ( ps' <-> [. G / h ]. [. h / f ]. ps )
29	25, 28	syl6bbr 267	. . . . . . 7 |- ( h = G -> ( [. h / f ]. ps <-> ps' ) )
30	19, 24, 29	3anbi123d 1341	. . . . . 6 |- ( h = G -> ( ( h Fn N /\ [. h / f ]. ph /\ [. h / f ]. ps ) <-> ( G Fn N /\ ph' /\ ps' ) ) )
31	30	anbi2d 711	. . . . 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 29520	. . . . 5 |- ( ( N e. D /\ G Fn N /\ ph' /\ ps' ) <-> ( N e. D /\ ( G Fn N /\ ph' /\ ps' ) ) )
33	31, 16, 32	3bitr4g 292	. . . 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 261	. . 3 |- ( h = G -> ( [. h / f ]. ch <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) ) )
35	3, 34	sbcie 3304	. 2 |- ( [. G / h ]. [. h / f ]. ch <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) )
36	1, 2, 35	3bitr2i 277	1 |- ( ch' <-> ( N e. D /\ G Fn N /\ ph' /\ ps' ) )

Syntax hints:   <-> wb 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   _Vcvv 3047   [.wsbc 3269   Fn wfn 5580   /\ w-bnj17 29503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-fun 5587  df-fn 5588  df-bnj17 29504

This theorem is referenced by:  bnj910  29771  bnj999  29780  bnj907  29788