Theorem eqsbc3rVD 41172

Description: Virtual deduction proof of eqsbc3r 3836. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eqsbc3rVD	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eqsbc3rVD

Step	Hyp	Ref	Expression
1		idn1 40906	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		eqsbc3 3816	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶))
3	1, 2	e1a 40959	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶)   )
4		eqcom 2828	. . . . . . . . . 10 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶)
5	4	sbcbii 3828	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶)
6	5	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶))
7	1, 6	e1a 40959	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶)   )
8		idn2 40945	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
9		biimp 217	. . . . . . 7 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥 → [𝐴 / 𝑥]𝑥 = 𝐶))
10	7, 8, 9	e12 41056	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
11		biimp 217	. . . . . 6 ⊢ (([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶 → 𝐴 = 𝐶))
12	3, 10, 11	e12 41056	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
13		eqcom 2828	. . . . 5 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
14	12, 13	e2bi 40964	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
15	14	in2 40937	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 → 𝐶 = 𝐴)   )
16		idn2 40945	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
17	16, 13	e2bir 40965	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
18		biimpr 222	. . . . . 6 ⊢ (([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶) → (𝐴 = 𝐶 → [𝐴 / 𝑥]𝑥 = 𝐶))
19	3, 17, 18	e12 41056	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
20		biimpr 222	. . . . 5 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶 → [𝐴 / 𝑥]𝐶 = 𝑥))
21	7, 19, 20	e12 41056	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
22	21	in2 40937	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 = 𝐴 → [𝐴 / 𝑥]𝐶 = 𝑥)   )
23		impbi 210	. . 3 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 → 𝐶 = 𝐴) → ((𝐶 = 𝐴 → [𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)))
24	15, 22, 23	e11 41020	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)   )
25	24	in1 40903	1 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  [wsbc 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772  df-vd1 40902  df-vd2 40910

This theorem is referenced by: (None)