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| 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 1795 | . 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑 | 
| 3 | mosubopt 5514 | . 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 1537 = wceq 1539 ∃wex 1778 ∃*wmo 2537 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 | 
| This theorem is referenced by: ov3 7597 ov6g 7598 oprabex3 8003 axaddf 11186 axmulf 11187 | 
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