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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.6 | Structured version Visualization version GIF version |
Description: Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
pm11.6 | ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2029 | . . 3 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
2 | an32 835 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓)) | |
3 | 2 | 2exbii 1765 | . . 3 ⊢ (∃𝑦∃𝑥((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓)) |
4 | 1, 3 | bitri 263 | . 2 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓)) |
5 | 19.41v 1901 | . . 3 ⊢ (∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒)) | |
6 | 5 | exbii 1764 | . 2 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒)) |
7 | 19.41v 1901 | . . 3 ⊢ (∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ (∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓)) | |
8 | 7 | exbii 1764 | . 2 ⊢ (∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓)) |
9 | 4, 6, 8 | 3bitr3i 289 | 1 ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-11 2021 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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