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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm10.56 | Structured version Visualization version GIF version |
Description: Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm10.56 | ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.45 625 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒))) | |
2 | 1 | aleximi 1838 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜓 ∧ 𝜒))) |
3 | 2 | imp 410 | 1 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 |
This theorem is referenced by: (None) |
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