Theorem eqsbc3rVD 39075

Description: Virtual deduction proof of eqsbc3r 3492. (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 38790	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		eqsbc3 3475	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶))
3	1, 2	e1a 38852	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶)   )
4		eqcom 2629	. . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶)
5	4	sbcbiiOLD 38741	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶))
6	1, 5	e1a 38852	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶)   )
7		idn2 38838	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
8		biimp 205	. . . . . . 7 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥 → [𝐴 / 𝑥]𝑥 = 𝐶))
9	6, 7, 8	e12 38951	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
10		biimp 205	. . . . . 6 ⊢ (([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶 → 𝐴 = 𝐶))
11	3, 9, 10	e12 38951	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
12		eqcom 2629	. . . . 5 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
13	11, 12	e2bi 38857	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
14	13	in2 38830	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 → 𝐶 = 𝐴)   )
15		idn2 38838	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
16	15, 12	e2bir 38858	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
17		biimpr 210	. . . . . 6 ⊢ (([𝐴 / 𝑥]𝑥 = 𝐶 ↔ 𝐴 = 𝐶) → (𝐴 = 𝐶 → [𝐴 / 𝑥]𝑥 = 𝐶))
18	3, 16, 17	e12 38951	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
19		biimpr 210	. . . . 5 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶 → [𝐴 / 𝑥]𝐶 = 𝑥))
20	6, 18, 19	e12 38951	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
21	20	in2 38830	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 = 𝐴 → [𝐴 / 𝑥]𝐶 = 𝑥)   )
22		impbi 198	. . 3 ⊢ (([𝐴 / 𝑥]𝐶 = 𝑥 → 𝐶 = 𝐴) → ((𝐶 = 𝐴 → [𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)))
23	14, 21, 22	e11 38913	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴)   )
24	23	in1 38787	1 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥 ↔ 𝐶 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)