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Mirrors > Home > MPE Home > Th. List > 19.43 | Structured version Visualization version GIF version |
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |
Ref | Expression |
---|---|
19.43 | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-or 844 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
2 | 1 | exbii 1848 | . . 3 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ∃𝑥(¬ 𝜑 → 𝜓)) |
3 | 19.35 1878 | . . 3 ⊢ (∃𝑥(¬ 𝜑 → 𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓)) | |
4 | alnex 1782 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | 4 | imbi1i 352 | . . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) |
6 | 2, 3, 5 | 3bitri 299 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) |
7 | df-or 844 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) | |
8 | 6, 7 | bitr4i 280 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 |
This theorem is referenced by: 19.34 1993 19.44v 1999 19.45v 2000 19.44 2239 19.45 2240 rexun 4166 unipr 4855 uniun 4861 unopab 5145 zfpair 5322 dmun 5779 dmopab2rex 5786 coundi 6100 coundir 6101 kmlem16 9591 vdwapun 16310 satfdm 32616 satf0op 32624 dmopab3rexdif 32652 bj-nnfor 34079 bj-nnford 34080 pm10.42 40716 |
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