Theorem preqr1 4379

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)

Hypotheses

Ref	Expression
preqr1.a	⊢ 𝐴 ∈ V
preqr1.b	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
preqr1	⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Proof of Theorem preqr1

Step	Hyp	Ref	Expression
1		preqr1.a	. . 3 ⊢ 𝐴 ∈ V
2		id 22	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		preqr1.b	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
5	2, 4	preq1b 4377	. . 3 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
6	1, 5	ax-mp 5	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)
7	6	biimpi 206	1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {cpr 4179

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-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  preqr2  4381  opthwiener  4976  cusgrfilem2  26352  usgr2wlkneq  26652  wopprc  37597