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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 3607 | . 2 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
2 | mosub.1 | . . 3 ⊢ ∃*𝑥𝜑 | |
3 | 2 | ax-gen 1802 | . 2 ⊢ ∀𝑦∃*𝑥𝜑 |
4 | moexexvw 2631 | . 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 399 ∀wal 1540 = wceq 1542 ∃wex 1786 ∃*wmo 2538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-mo 2540 df-cleq 2730 |
This theorem is referenced by: (None) |
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