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| Mirrors > Home > MPE Home > Th. List > mosub | Structured version Visualization version GIF version | ||
| Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
| Ref | Expression |
|---|---|
| mosub.1 | ⊢ ∃*𝑥𝜑 |
| Ref | Expression |
|---|---|
| mosub | ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | moeq 3713 | . 2 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
| 2 | mosub.1 | . . 3 ⊢ ∃*𝑥𝜑 | |
| 3 | 2 | ax-gen 1795 | . 2 ⊢ ∀𝑦∃*𝑥𝜑 |
| 4 | moexexvw 2628 | . 2 ⊢ ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-cleq 2729 |
| This theorem is referenced by: (None) |
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