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Mirrors > Home > MPE Home > Th. List > 19.43OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of 19.43 1890. Do not delete as it is referenced on the mmrecent.html 1890 page and in conventions-labels 28438. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
19.43OLD | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioran 984 | . . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) | |
2 | 1 | albii 1827 | . . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 ∧ ¬ 𝜓)) |
3 | 19.26 1878 | . . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓)) | |
4 | alnex 1789 | . . . . 5 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | alnex 1789 | . . . . 5 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
6 | 4, 5 | anbi12i 630 | . . . 4 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓)) |
7 | 2, 3, 6 | 3bitri 300 | . . 3 ⊢ (∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓)) |
8 | 7 | notbii 323 | . 2 ⊢ (¬ ∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓)) |
9 | df-ex 1788 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ¬ ∀𝑥 ¬ (𝜑 ∨ 𝜓)) | |
10 | oran 990 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓)) | |
11 | 8, 9, 10 | 3bitr4i 306 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 |
This theorem is referenced by: (None) |
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