Theorem preq2b 4801

Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.)

Hypotheses

Ref	Expression
preq1b.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
preq1b.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
preq2b	⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem preq2b

Step	Hyp	Ref	Expression
1		prcom 4687	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4687	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2752	. 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4		preq1b.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		preq1b.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	4, 5	preq1b 4800	. 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
7	3, 6	bitrid 283	1 ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {cpr 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-sn 4579  df-pr 4581

This theorem is referenced by:  umgr2v2enb1  29549  clsk1indlem4  44227  usgrexmpl2nb0  48219  usgrexmpl2nb1  48220  usgrexmpl2nb2  48221  usgrexmpl2nb3  48222  usgrexmpl2nb4  48223  usgrexmpl2nb5  48224