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Theorem rmoanimALT 3827
 Description: Alternate proof of rmoanim 3826, shorter but requiring ax-10 2143 and ax-11 2159. (Contributed by Alexander van der Vekens, 25-Jun-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
rmoanim.1 𝑥𝜑
Assertion
Ref Expression
rmoanimALT (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoanimALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 impexp 454 . . . . 5 (((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓𝑥 = 𝑦)))
21ralbii 3136 . . . 4 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)))
3 rmoanim.1 . . . . 5 𝑥𝜑
43r19.21 3182 . . . 4 (∀𝑥𝐴 (𝜑 → (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
52, 4bitri 278 . . 3 (∀𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
65exbii 1849 . 2 (∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
7 nfv 1915 . . 3 𝑦(𝜑𝜓)
87rmo2 3819 . 2 (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃𝑦𝑥𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))
9 nfv 1915 . . . . 5 𝑦𝜓
109rmo2 3819 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦))
1110imbi2i 339 . . 3 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 19.37v 1998 . . 3 (∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦𝑥𝐴 (𝜓𝑥 = 𝑦)))
1311, 12bitr4i 281 . 2 ((𝜑 → ∃*𝑥𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
146, 8, 133bitr4i 306 1 (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785  ∀wral 3109  ∃*wrmo 3112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2601  df-ral 3114  df-rmo 3117 This theorem is referenced by: (None)
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