Theorem preq2b 4823

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 4708	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4708	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2753	. 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 4822	. 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 1540   ∈ wcel 2108  {cpr 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-sn 4602  df-pr 4604

This theorem is referenced by:  umgr2v2enb1  29506  clsk1indlem4  44068  usgrexmpl2nb0  48035  usgrexmpl2nb1  48036  usgrexmpl2nb2  48037  usgrexmpl2nb3  48038  usgrexmpl2nb4  48039  usgrexmpl2nb5  48040