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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 4267 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
2 | prcom 4267 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
3 | 1, 2 | eqeq12i 2636 | . 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 4377 | . 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 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: umgr2v2enb1 26422 clsk1indlem4 38342 |
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