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Mirrors > Home > MPE Home > Th. List > r19.29vva | Structured version Visualization version GIF version |
Description: A commonly used pattern based on r19.29 3251, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) |
Ref | Expression |
---|---|
r19.29vva.1 | ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) |
r19.29vva.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
r19.29vva | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29vva.1 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 402 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | ralrimiva 3145 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒)) |
4 | 3 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒)) |
5 | r19.29vva.2 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
6 | 4, 5 | r19.29d2r 3259 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓)) |
7 | pm3.35 838 | . . . . 5 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒) | |
8 | 7 | ancoms 451 | . . . 4 ⊢ (((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
9 | 8 | rexlimivw 3208 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
10 | 9 | rexlimivw 3208 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
11 | 6, 10 | syl 17 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∀wral 3087 ∃wrex 3088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-ral 3092 df-rex 3093 |
This theorem is referenced by: trust 22358 utoptop 22363 metustto 22683 restmetu 22700 tgbtwndiff 25750 legov 25829 legso 25843 tglnne 25872 tglndim0 25873 tglinethru 25880 tglnne0 25884 tglnpt2 25885 footex 25962 midex 25978 opptgdim2 25986 cgrane1 26053 cgrane2 26054 cgrane3 26055 cgrane4 26056 cgrahl1 26057 cgrahl2 26058 cgracgr 26059 cgratr 26064 cgrabtwn 26066 cgrahl 26067 dfcgra2 26070 sacgr 26071 acopyeu 26074 f1otrge 26101 archiabllem2c 30257 txomap 30409 qtophaus 30411 pstmfval 30447 eulerpartlemgvv 30946 tgoldbachgtd 31252 irrapxlem4 38163 |
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