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Mirrors > Home > MPE Home > Th. List > 2rexuz | Structured version Visualization version GIF version |
Description: Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.) |
Ref | Expression |
---|---|
2rexuz | ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexuz2 11615 | . . 3 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ (𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))) | |
2 | 1 | exbii 1764 | . 2 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))) |
3 | df-rex 2902 | . 2 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑) ↔ ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))) | |
4 | 2, 3 | bitr4i 266 | 1 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 ≤ cle 9954 ℤcz 11254 ℤ≥cuz 11563 |
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-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-neg 10148 df-z 11255 df-uz 11564 |
This theorem is referenced by: (None) |
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