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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 4407 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
2 | prcom 4407 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
3 | 1, 2 | eqeq12i 2770 | . 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 4518 | . 2 ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)) |
7 | 3, 6 | syl5bb 272 | 1 ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1628 ∈ wcel 2135 {cpr 4319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-v 3338 df-un 3716 df-sn 4318 df-pr 4320 |
This theorem is referenced by: umgr2v2enb1 26628 clsk1indlem4 38840 |
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