Theorem preq1b 4377

Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first 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
preq1b	⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Proof of Theorem preq1b

Step	Hyp	Ref	Expression
1		preq1b.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		prid1g 4295	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐶})
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴, 𝐶})
4		eleq2 2690	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
5	3, 4	syl5ibcom 235	. . . . . 6 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
6		elprg 4196	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
8	5, 7	sylibd 229	. . . . 5 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
9	8	imp 445	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
10		preq1b.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		prid1g 4295	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
13		eleq2 2690	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
14	12, 13	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
15		elprg 4196	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
16	10, 15	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
17	14, 16	sylibd 229	. . . . 5 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
18	17	imp 445	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
19		eqcom 2629	. . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
20		eqeq2 2633	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
21	9, 18, 19, 20	oplem1 1007	. . 3 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
22	21	ex 450	. 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
23		preq1 4268	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
24	22, 23	impbid1 215	1 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {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:  preq2b  4378  preqr1  4379  preqr1g  4385  uhgr3cyclexlem  27041