Theorem eqsbc3rVD 16664

Description: Virtual deduction proof of eqsbc3r 2507.

Assertion

Ref	Expression
eqsbc3rVD	|- (A e. B -> ([A / x]C = x <-> C = A))

Distinct variable groups:   x,C   x,A

Proof of Theorem eqsbc3rVD

Step	Hyp	Ref	Expression
1		idn1 16484	. . . . . . 7 |- . A e. B   ⊢   A e. B .
2		eqsbc3 2494	. . . . . . 7 |- (A e. B -> ([A / x]x = C <-> A = C))
3	1, 2	e1_ 16518	. . . . . 6 |- . A e. B   ⊢   ([A / x]x = C <-> A = C) .
4		eqcom 1886	. . . . . . . . 9 |- (C = x <-> x = C)
5	4	sbcbii 2506	. . . . . . . 8 |- (A e. B -> ([A / x]C = x <-> [A / x]x = C))
6	1, 5	e1_ 16518	. . . . . . 7 |- . A e. B   ⊢   ([A / x]C = x <-> [A / x]x = C) .
7		idn2 16509	. . . . . . 7 |- . A e. B, [A / x]C = x   ⊢   [A / x]C = x .
8		bi1 165	. . . . . . 7 |- (([A / x]C = x <-> [A / x]x = C) -> ([A / x]C = x -> [A / x]x = C))
9	6, 7, 8	e12 16593	. . . . . 6 |- . A e. B, [A / x]C = x   ⊢   [A / x]x = C .
10		bi1 165	. . . . . 6 |- (([A / x]x = C <-> A = C) -> ([A / x]x = C -> A = C))
11	3, 9, 10	e12 16593	. . . . 5 |- . A e. B, [A / x]C = x   ⊢   A = C .
12		eqcom 1886	. . . . 5 |- (A = C <-> C = A)
13	11, 12	e2bi 16522	. . . 4 |- . A e. B, [A / x]C = x   ⊢   C = A .
14	13	in2 16506	. . 3 |- . A e. B   ⊢   ([A / x]C = x -> C = A) .
15		idn2 16509	. . . . . . 7 |- . A e. B, C = A   ⊢   C = A .
16	15, 12	e2bir 16523	. . . . . 6 |- . A e. B, C = A   ⊢   A = C .
17		bi2 166	. . . . . 6 |- (([A / x]x = C <-> A = C) -> (A = C -> [A / x]x = C))
18	3, 16, 17	e12 16593	. . . . 5 |- . A e. B, C = A   ⊢   [A / x]x = C .
19		bi2 166	. . . . 5 |- (([A / x]C = x <-> [A / x]x = C) -> ([A / x]x = C -> [A / x]C = x))
20	6, 18, 19	e12 16593	. . . 4 |- . A e. B, C = A   ⊢   [A / x]C = x .
21	20	in2 16506	. . 3 |- . A e. B   ⊢   (C = A -> [A / x]C = x) .
22		bi3 167	. . 3 |- (([A / x]C = x -> C = A) -> ((C = A -> [A / x]C = x) -> ([A / x]C = x <-> C = A)))
23	14, 21, 22	e11 16578	. 2 |- . A e. B   ⊢   ([A / x]C = x <-> C = A) .
24	23	in1 16481	1 |- (A e. B -> ([A / x]C = x <-> C = A))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-vd1 16480  df-vd2 16489

Copyright terms: Public domain