Theorem eqvinc 1407

Description: A variable introduction law for class equality.

Hypothesis

Ref	Expression
eqvinc.1	⊢ A ∈ V

Assertion

Ref	Expression
eqvinc	⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. . 3 ⊢ A ∈ V
2		eleq1 1149	. . 3 ⊢ (A = B → (A ∈ V ↔ B ∈ V))
3	1, 2	mpbii 168	. 2 ⊢ (A = B → B ∈ V)
4		visset 1350	. . . . 5 ⊢ x ∈ V
5		eleq1 1149	. . . . 5 ⊢ (x = B → (x ∈ V ↔ B ∈ V))
6	4, 5	mpbii 168	. . . 4 ⊢ (x = B → B ∈ V)
7	6	adantl 305	. . 3 ⊢ ((x = A ∧ x = B) → B ∈ V)
8	7	19.23aiv 952	. 2 ⊢ (∃x(x = A ∧ x = B) → B ∈ V)
9		cleq2 1110	. . 3 ⊢ (z = B → (A = z ↔ A = B))
10		cleq2 1110	. . . . 5 ⊢ (z = B → (x = z ↔ x = B))
11	10	anbi2d 468	. . . 4 ⊢ (z = B → ((x = A ∧ x = z) ↔ (x = A ∧ x = B)))
12	11	biexdv 936	. . 3 ⊢ (z = B → (∃x(x = A ∧ x = z) ↔ ∃x(x = A ∧ x = B)))
13		cleq1 1107	. . . 4 ⊢ (y = A → (y = z ↔ A = z))
14		cleq1 1107	. . . . . . 7 ⊢ (y = A → (y = x ↔ A = x))
15		cleqcom 1103	. . . . . . 7 ⊢ (A = x ↔ x = A)
16	14, 15	syl6bb 414	. . . . . 6 ⊢ (y = A → (y = x ↔ x = A))
17	16	anbi1d 469	. . . . 5 ⊢ (y = A → ((y = x ∧ x = z) ↔ (x = A ∧ x = z)))
18	17	biexdv 936	. . . 4 ⊢ (y = A → (∃x(y = x ∧ x = z) ↔ ∃x(x = A ∧ x = z)))
19		eqvin 932	. . . 4 ⊢ (y = z ↔ ∃x(y = x ∧ x = z))
20	1, 13, 18, 19	vtoclb 1381	. . 3 ⊢ (A = z ↔ ∃x(x = A ∧ x = z))
21	9, 12, 20	vtoclbg 1384	. 2 ⊢ (B ∈ V → (A = B ↔ ∃x(x = A ∧ x = B)))
22	3, 8, 21	pm5.21nii 504	1 ⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  eqvincf 1408  opabid 2099  findsg 2398  tfindsg 2402  f1fv 2916  indpi 3828

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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