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Theorem rmoanim 3877
Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2701. (Contributed by Alexander van der Vekens, 25-Jun-2017.) Avoid ax-10 2141 and ax-11 2157. (Revised by Gino Giotto, 24-Aug-2023.)
Hypothesis
Ref Expression
rmoanim.1 𝑥𝜑
Assertion
Ref Expression
rmoanim (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoanim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 impexp 453 . . . . 5 (((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓𝑥 = 𝑦)))
21ralbii 3165 . . . 4 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)))
3 rmoanim.1 . . . . 5 𝑥𝜑
43r19.21 3215 . . . 4 (∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
52, 4bitri 277 . . 3 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
65exbii 1844 . 2 (∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
7 df-rmo 3146 . . 3 (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
8 df-mo 2618 . . 3 (∃*𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ ∃𝑦𝑥((𝑥𝐴 ∧ (𝜑𝜓)) → 𝑥 = 𝑦))
9 impexp 453 . . . . . 6 (((𝑥𝐴 ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)))
109albii 1816 . . . . 5 (∀𝑥((𝑥𝐴 ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)))
11 df-ral 3143 . . . . 5 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → ((𝜑𝜓) → 𝑥 = 𝑦)))
1210, 11bitr4i 280 . . . 4 (∀𝑥((𝑥𝐴 ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
1312exbii 1844 . . 3 (∃𝑦𝑥((𝑥𝐴 ∧ (𝜑𝜓)) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
147, 8, 133bitri 299 . 2 (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
15 df-rmo 3146 . . . . 5 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
16 df-mo 2618 . . . . 5 (∃*𝑥(𝑥𝐴𝜓) ↔ ∃𝑦𝑥((𝑥𝐴𝜓) → 𝑥 = 𝑦))
17 impexp 453 . . . . . . . 8 (((𝑥𝐴𝜓) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜓𝑥 = 𝑦)))
1817albii 1816 . . . . . . 7 (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝑥 = 𝑦)))
19 df-ral 3143 . . . . . . 7 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝑥 = 𝑦)))
2018, 19bitr4i 280 . . . . . 6 (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
2120exbii 1844 . . . . 5 (∃𝑦𝑥((𝑥𝐴𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦))
2215, 16, 213bitri 299 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦))
2322imbi2i 338 . . 3 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
24 19.37v 1994 . . 3 (∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
2523, 24bitr4i 280 . 2 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
266, 14, 253bitr4i 305 1 (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1531  wex 1776  wnf 1780  wcel 2110  ∃*wmo 2616  wral 3138  ∃*wrmo 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-mo 2618  df-ral 3143  df-rmo 3146
This theorem is referenced by:  2reu1  3880
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