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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 3695 | . 2 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
2 | mosub.1 | . . 3 ⊢ ∃*𝑥𝜑 | |
3 | 2 | ax-gen 1787 | . 2 ⊢ ∀𝑦∃*𝑥𝜑 |
4 | moexexvw 2706 | . 2 ⊢ ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) | |
5 | 1, 3, 4 | mp2an 688 | 1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∃*wmo 2613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-cleq 2811 |
This theorem is referenced by: (None) |
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