Theorem preq1b 4771

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 4690	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐶})
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴, 𝐶})
4		eleq2 2901	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
5	3, 4	syl5ibcom 247	. . . . . 6 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
6		elprg 4582	. . . . . . 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 241	. . . . 5 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
9	8	imp 409	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
10		preq1b.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		prid1g 4690	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
13		eleq2 2901	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
14	12, 13	syl5ibrcom 249	. . . . . 6 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
15		elprg 4582	. . . . . . 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 241	. . . . 5 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
18	17	imp 409	. . . 4 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
19		eqcom 2828	. . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
20		eqeq2 2833	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
21	9, 18, 19, 20	oplem1 1051	. . 3 ⊢ ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
22	21	ex 415	. 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
23		preq1 4663	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
24	22, 23	impbid1 227	1 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-sn 4562  df-pr 4564

This theorem is referenced by:  preq2b  4772  preqr1  4773  preqr1g  4777  uhgr3cyclexlem  27954