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Theorem 2rmoswap 3756
Description: A condition allowing to swap restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2728. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2rmoswap (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2rmoswap
StepHypRef Expression
1 df-rmo 3151 . . 3 (∃*𝑦𝐵 𝜑 ↔ ∃*𝑦(𝑦𝐵𝜑))
21ralbii 3170 . 2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑))
3 df-ral 3148 . . . 4 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
4 moanimv 2703 . . . . 5 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
54albii 1813 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
63, 5bitr4i 279 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 2moswapv 2713 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃*𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃*𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑))))
8 df-rmo 3151 . . . . 5 (∃*𝑥𝐴𝑦𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
9 r19.42v 3355 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
10 df-rex 3149 . . . . . . . 8 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
11 an12 641 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1211exbii 1841 . . . . . . . 8 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1310, 12bitri 276 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
149, 13bitr3i 278 . . . . . 6 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1514mobii 2629 . . . . 5 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃*𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
168, 15bitri 276 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 ↔ ∃*𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
17 df-rmo 3151 . . . . 5 (∃*𝑦𝐵𝑥𝐴 𝜑 ↔ ∃*𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
18 r19.42v 3355 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
19 df-rex 3149 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2018, 19bitr3i 278 . . . . . 6 ((𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2120mobii 2629 . . . . 5 (∃*𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃*𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2217, 21bitri 276 . . . 4 (∃*𝑦𝐵𝑥𝐴 𝜑 ↔ ∃*𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
237, 16, 223imtr4g 297 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))
246, 23sylbi 218 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))
252, 24sylbi 218 1 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773  wcel 2107  ∃*wmo 2618  wral 3143  wrex 3144  ∃*wrmo 3146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-ral 3148  df-rex 3149  df-rmo 3151
This theorem is referenced by:  2reu1  3885
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