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Mirrors > Home > MPE Home > Th. List > Mathboxes > bropabid | Structured version Visualization version GIF version |
Description: Lemma for ~? inxptxp . (Contributed by Peter Mazsa, 24-Nov-2018.) |
Ref | Expression |
---|---|
bropabid.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
bropabid | ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bropabid.1 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1 | breqi 4659 | . 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦) |
3 | df-br 4654 | . 2 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | opabid 4982 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
5 | 2, 3, 4 | 3bitri 286 | 1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 {copab 4712 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 |
This theorem is referenced by: (None) |
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