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Mirrors > Home > MPE Home > Th. List > eeorOLD | Structured version Visualization version GIF version |
Description: Obsolete version of eeor 2323 as of 21-Nov-2024. (Contributed by NM, 8-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eeor.1 | ⊢ Ⅎ𝑦𝜑 |
eeor.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
eeorOLD | ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeor.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.45 2223 | . . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓)) |
3 | 2 | exbii 1842 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓)) |
4 | eeor.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfex 2311 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
6 | 5 | 19.44 2222 | . 2 ⊢ (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
7 | 3, 6 | bitri 275 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1774 df-nf 1778 |
This theorem is referenced by: (None) |
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