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Mirrors > Home > MPE Home > Th. List > rexuz | Structured version Visualization version GIF version |
Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
Ref | Expression |
---|---|
rexuz | ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz1 12250 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))) | |
2 | 1 | anbi1d 631 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ∧ 𝜑))) |
3 | anass 471 | . . 3 ⊢ (((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑))) | |
4 | 2, 3 | syl6bb 289 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) |
5 | 4 | rexbidv2 3298 | 1 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ∃wrex 3142 class class class wbr 5069 ‘cfv 6358 ≤ cle 10679 ℤcz 11984 ℤ≥cuz 12246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-neg 10876 df-z 11985 df-uz 12247 |
This theorem is referenced by: (None) |
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