Theorem preqr2g 4127

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
preqr2g	⊢ ((A ∈ V ∧ B ∈ W) → ({C, A} = {C, B} → A = B))

Proof of Theorem preqr2g

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

Step	Hyp	Ref	Expression
1		preq2 3801	. . . 4 ⊢ (x = A → {C, x} = {C, A})
2	1	eqeq1d 2361	. . 3 ⊢ (x = A → ({C, x} = {C, y} ↔ {C, A} = {C, y}))
3		eqeq1 2359	. . 3 ⊢ (x = A → (x = y ↔ A = y))
4	2, 3	imbi12d 311	. 2 ⊢ (x = A → (({C, x} = {C, y} → x = y) ↔ ({C, A} = {C, y} → A = y)))
5		preq2 3801	. . . 4 ⊢ (y = B → {C, y} = {C, B})
6	5	eqeq2d 2364	. . 3 ⊢ (y = B → ({C, A} = {C, y} ↔ {C, A} = {C, B}))
7		eqeq2 2362	. . 3 ⊢ (y = B → (A = y ↔ A = B))
8	6, 7	imbi12d 311	. 2 ⊢ (y = B → (({C, A} = {C, y} → A = y) ↔ ({C, A} = {C, B} → A = B)))
9		vex 2863	. . 3 ⊢ x ∈ V
10		vex 2863	. . 3 ⊢ y ∈ V
11	9, 10	preqr2 4126	. 2 ⊢ ({C, x} = {C, y} → x = y)
12	4, 8, 11	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → ({C, A} = {C, B} → A = B))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  opkthg  4132