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Theorem rmoanim 39828
 Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2517. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
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 461 . . . . 5 (((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓𝑥 = 𝑦)))
21ralbii 2963 . . . 4 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)))
3 rmoanim.1 . . . . 5 𝑥𝜑
43r19.21 2939 . . . 4 (∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
52, 4bitri 263 . . 3 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
65exbii 1764 . 2 (∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
7 nfv 1830 . . 3 𝑦(𝜑𝜓)
87rmo2 3492 . 2 (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
9 nfv 1830 . . . . 5 𝑦𝜓
109rmo2 3492 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦))
1110imbi2i 325 . . 3 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 19.37v 1897 . . 3 (∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
1311, 12bitr4i 266 . 2 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
146, 8, 133bitr4i 291 1 (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699  ∀wral 2896  ∃*wrmo 2899 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ral 2901  df-rmo 2904 This theorem is referenced by:  2reu1  39835
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