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| Mirrors > Home > MPE Home > Th. List > preq2b | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| preq1b.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| preq1b.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| Ref | Expression | 
|---|---|
| preq2b | ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | prcom 4732 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
| 2 | prcom 4732 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
| 3 | 1, 2 | eqeq12i 2755 | . 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 4846 | . 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)) | 
| 7 | 3, 6 | bitrid 283 | 1 ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cpr 4628 | 
| 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 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: umgr2v2enb1 29544 clsk1indlem4 44057 usgrexmpl2nb0 47990 usgrexmpl2nb1 47991 usgrexmpl2nb2 47992 usgrexmpl2nb3 47993 usgrexmpl2nb4 47994 usgrexmpl2nb5 47995 | 
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