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Theorem reuan 3828
Description: Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2686. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Hypothesis
Ref Expression
rmoanim.1 𝑥𝜑
Assertion
Ref Expression
reuan (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reuan
StepHypRef Expression
1 rmoanim.1 . . . . . 6 𝑥𝜑
2 simpl 486 . . . . . . 7 ((𝜑𝜓) → 𝜑)
32a1i 11 . . . . . 6 (𝑥𝐴 → ((𝜑𝜓) → 𝜑))
41, 3rexlimi 3277 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → 𝜑)
54adantr 484 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → 𝜑)
6 simpr 488 . . . . . 6 ((𝜑𝜓) → 𝜓)
76reximi 3209 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
87adantr 484 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
9 nfre1 3268 . . . . . 6 𝑥𝑥𝐴 (𝜑𝜓)
104adantr 484 . . . . . . . . 9 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → 𝜑)
1110a1d 25 . . . . . . . 8 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓𝜑))
1211ancrd 555 . . . . . . 7 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓 → (𝜑𝜓)))
136, 12impbid2 229 . . . . . 6 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → ((𝜑𝜓) ↔ 𝜓))
149, 13rmobida 3348 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃*𝑥𝐴 𝜓))
1514biimpa 480 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃*𝑥𝐴 𝜓)
165, 8, 15jca32 519 . . 3 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
17 reu5 3378 . . 3 (∃!𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)))
18 reu5 3378 . . . 4 (∃!𝑥𝐴 𝜓 ↔ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓))
1918anbi2i 625 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) ↔ (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
2016, 17, 193imtr4i 295 . 2 (∃!𝑥𝐴 (𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝐴 𝜓))
21 ibar 532 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2221adantr 484 . . . 4 ((𝜑𝑥𝐴) → (𝜓 ↔ (𝜑𝜓)))
231, 22reubida 3343 . . 3 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 (𝜑𝜓)))
2423biimpa 480 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 (𝜑𝜓))
2520, 24impbii 212 1 (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wnf 1785  wcel 2112  wrex 3110  ∃!wreu 3111  ∃*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-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2601  df-eu 2632  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117
This theorem is referenced by:  2reu7  43654  2reu8  43655
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