Theorem preq2b 4814

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 4700	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4700	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2783	. 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 4813	. 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
7	3, 6	bitrid 286	1 ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {cpr 4593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-sn 4592  df-pr 4594

This theorem is used by:  umgr2v2enb1  29938  clsk1indlem4  44847  usgrexmpl2nb0  48873  usgrexmpl2nb1  48874  usgrexmpl2nb2  48875  usgrexmpl2nb3  48876  usgrexmpl2nb4  48877  usgrexmpl2nb5  48878