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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 847 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
2 | 1 | exbii 1851 | . . 3 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ∃𝑥(¬ 𝜑 → 𝜓)) |
3 | 19.35 1881 | . . 3 ⊢ (∃𝑥(¬ 𝜑 → 𝜓) ↔ (∀𝑥 ¬ 𝜑 → ∃𝑥𝜓)) | |
4 | alnex 1784 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | 4 | imbi1i 350 | . . 3 ⊢ ((∀𝑥 ¬ 𝜑 → ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) |
6 | 2, 3, 5 | 3bitri 297 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) |
7 | df-or 847 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 → ∃𝑥𝜓)) | |
8 | 6, 7 | bitr4i 278 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 |
This theorem is referenced by: 19.34 1991 19.44v 1997 19.45v 1998 19.44 2231 19.45 2232 eeor 2330 rexun 4191 uniprg 4926 uniprOLD 4928 uniun 4935 unopab 5231 zfpair 5420 dmun 5911 dmopab2rex 5918 coundi 6247 coundir 6248 kmlem16 10160 vdwapun 16907 satfdm 34360 satf0op 34368 dmopab3rexdif 34396 bj-nnfor 35628 bj-nnford 35629 pm10.42 43123 |
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