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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 1793 | . 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑 |
3 | mosubopt 5520 | . 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 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∃*wmo 2536 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: ov3 7596 ov6g 7597 oprabex3 8001 axaddf 11183 axmulf 11184 |
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