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Mirrors > Home > MPE Home > Th. List > mosubop | Structured version Visualization version GIF version |
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
mosubop.1 | ⊢ ∃*𝑥𝜑 |
Ref | Expression |
---|---|
mosubop | ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mosubop.1 | . . 3 ⊢ ∃*𝑥𝜑 | |
2 | 1 | gen2 1723 | . 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑 |
3 | mosubopt 4972 | . 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑)) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 ∃*wmo 2471 〈cop 4183 |
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 |
This theorem is referenced by: ov3 6797 ov6g 6798 oprabex3 7157 axaddf 9966 axmulf 9967 |
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