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Theorem reuan 41697
Description: Introduction of a conjunct into restricted unique existential quantifier, analogous to euan 2705. (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 470 . . . . . . 7 ((𝜑𝜓) → 𝜑)
32a1i 11 . . . . . 6 (𝑥𝐴 → ((𝜑𝜓) → 𝜑))
41, 3rexlimi 3223 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → 𝜑)
54adantr 468 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → 𝜑)
6 simpr 473 . . . . . 6 ((𝜑𝜓) → 𝜓)
76reximi 3209 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
87adantr 468 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
9 nfre1 3203 . . . . . 6 𝑥𝑥𝐴 (𝜑𝜓)
104adantr 468 . . . . . . . . 9 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → 𝜑)
1110a1d 25 . . . . . . . 8 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓𝜑))
1211ancrd 543 . . . . . . 7 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓 → (𝜑𝜓)))
136, 12impbid2 217 . . . . . 6 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → ((𝜑𝜓) ↔ 𝜓))
149, 13rmobida 3329 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃*𝑥𝐴 𝜓))
1514biimpa 464 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃*𝑥𝐴 𝜓)
165, 8, 15jca32 507 . . 3 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
17 reu5 3359 . . 3 (∃!𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)))
18 reu5 3359 . . . 4 (∃!𝑥𝐴 𝜓 ↔ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓))
1918anbi2i 611 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) ↔ (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
2016, 17, 193imtr4i 283 . 2 (∃!𝑥𝐴 (𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝐴 𝜓))
21 ibar 520 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2221adantr 468 . . . 4 ((𝜑𝑥𝐴) → (𝜓 ↔ (𝜑𝜓)))
231, 22reubida 3324 . . 3 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 (𝜑𝜓)))
2423biimpa 464 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 (𝜑𝜓))
2520, 24impbii 200 1 (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wnf 1863  wcel 2157  wrex 3108  ∃!wreu 3109  ∃*wrmo 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115
This theorem is referenced by:  2reu7  41708  2reu8  41709
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