Theorem 2sbc6g 36809

Description: Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
2sbc6g	|- ( ( A e. C /\ B e. D ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )

Distinct variable groups:   z, w, A   w, B, z

Allowed substitution hints:   ph( z, w)   C( z, w)   D( z, w)

Proof of Theorem 2sbc6g

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2472	. . . . . . 7 |- ( y = B -> ( w = y <-> w = B ) )
2	1	anbi2d 715	. . . . . 6 |- ( y = B -> ( ( z = x /\ w = y ) <-> ( z = x /\ w = B ) ) )
3	2	imbi1d 323	. . . . 5 |- ( y = B -> ( ( ( z = x /\ w = y ) -> ph ) <-> ( ( z = x /\ w = B ) -> ph ) ) )
4	3	2albidv 1779	. . . 4 |- ( y = B -> ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> A. z A. w ( ( z = x /\ w = B ) -> ph ) ) )
5		dfsbcq 3280	. . . . 5 |- ( y = B -> ( [. y / w ]. ph <-> [. B / w ]. ph ) )
6	5	sbcbidv 3333	. . . 4 |- ( y = B -> ( [. x / z ]. [. y / w ]. ph <-> [. x / z ]. [. B / w ]. ph ) )
7	4, 6	bibi12d 327	. . 3 |- ( y = B -> ( ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> [. x / z ]. [. y / w ]. ph ) <-> ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> [. x / z ]. [. B / w ]. ph ) ) )
8		eqeq2 2472	. . . . . . 7 |- ( x = A -> ( z = x <-> z = A ) )
9	8	anbi1d 716	. . . . . 6 |- ( x = A -> ( ( z = x /\ w = B ) <-> ( z = A /\ w = B ) ) )
10	9	imbi1d 323	. . . . 5 |- ( x = A -> ( ( ( z = x /\ w = B ) -> ph ) <-> ( ( z = A /\ w = B ) -> ph ) ) )
11	10	2albidv 1779	. . . 4 |- ( x = A -> ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> A. z A. w ( ( z = A /\ w = B ) -> ph ) ) )
12		dfsbcq 3280	. . . 4 |- ( x = A -> ( [. x / z ]. [. B / w ]. ph <-> [. A / z ]. [. B / w ]. ph ) )
13	11, 12	bibi12d 327	. . 3 |- ( x = A -> ( ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> [. x / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) ) )
14		vex 3059	. . . . 5 |- x e. _V
15	14	sbc6 3305	. . . 4 |- ( [. x / z ]. [. y / w ]. ph <-> A. z ( z = x -> [. y / w ]. ph ) )
16		19.21v 1796	. . . . . 6 |- ( A. w ( z = x -> ( w = y -> ph ) ) <-> ( z = x -> A. w ( w = y -> ph ) ) )
17		impexp 452	. . . . . . 7 |- ( ( ( z = x /\ w = y ) -> ph ) <-> ( z = x -> ( w = y -> ph ) ) )
18	17	albii 1701	. . . . . 6 |- ( A. w ( ( z = x /\ w = y ) -> ph ) <-> A. w ( z = x -> ( w = y -> ph ) ) )
19		vex 3059	. . . . . . . 8 |- y e. _V
20	19	sbc6 3305	. . . . . . 7 |- ( [. y / w ]. ph <-> A. w ( w = y -> ph ) )
21	20	imbi2i 318	. . . . . 6 |- ( ( z = x -> [. y / w ]. ph ) <-> ( z = x -> A. w ( w = y -> ph ) ) )
22	16, 18, 21	3bitr4ri 286	. . . . 5 |- ( ( z = x -> [. y / w ]. ph ) <-> A. w ( ( z = x /\ w = y ) -> ph ) )
23	22	albii 1701	. . . 4 |- ( A. z ( z = x -> [. y / w ]. ph ) <-> A. z A. w ( ( z = x /\ w = y ) -> ph ) )
24	15, 23	bitr2i 258	. . 3 |- ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> [. x / z ]. [. y / w ]. ph )
25	7, 13, 24	vtocl2g 3122	. 2 |- ( ( B e. D /\ A e. C ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )
26	25	ancoms 459	1 |- ( ( A e. C /\ B e. D ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452   = wceq 1454   e. wcel 1897   [.wsbc 3278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279

This theorem is referenced by:  pm14.123a  36819