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Theorem reuan 40481
Description: Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2529. (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 473 . . . . . . 7 ((𝜑𝜓) → 𝜑)
32a1i 11 . . . . . 6 (𝑥𝐴 → ((𝜑𝜓) → 𝜑))
41, 3rexlimi 3017 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → 𝜑)
54adantr 481 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → 𝜑)
6 simpr 477 . . . . . 6 ((𝜑𝜓) → 𝜓)
76reximi 3005 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
87adantr 481 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃𝑥𝐴 𝜓)
9 nfre1 2999 . . . . . 6 𝑥𝑥𝐴 (𝜑𝜓)
104adantr 481 . . . . . . . . 9 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → 𝜑)
1110a1d 25 . . . . . . . 8 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓𝜑))
1211ancrd 576 . . . . . . 7 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜓 → (𝜑𝜓)))
136, 12impbid2 216 . . . . . 6 ((∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → ((𝜑𝜓) ↔ 𝜓))
149, 13rmobida 3118 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 (𝜑𝜓) ↔ ∃*𝑥𝐴 𝜓))
1514biimpa 501 . . . 4 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → ∃*𝑥𝐴 𝜓)
165, 8, 15jca32 557 . . 3 ((∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)) → (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
17 reu5 3148 . . 3 (∃!𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 (𝜑𝜓) ∧ ∃*𝑥𝐴 (𝜑𝜓)))
18 reu5 3148 . . . 4 (∃!𝑥𝐴 𝜓 ↔ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓))
1918anbi2i 729 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) ↔ (𝜑 ∧ (∃𝑥𝐴 𝜓 ∧ ∃*𝑥𝐴 𝜓)))
2016, 17, 193imtr4i 281 . 2 (∃!𝑥𝐴 (𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝐴 𝜓))
21 ibar 525 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2221adantr 481 . . . 4 ((𝜑𝑥𝐴) → (𝜓 ↔ (𝜑𝜓)))
231, 22reubida 3113 . . 3 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 (𝜑𝜓)))
2423biimpa 501 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 (𝜑𝜓))
2520, 24impbii 199 1 (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wnf 1705  wcel 1987  wrex 2908  ∃!wreu 2909  ∃*wrmo 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-eu 2473  df-mo 2474  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915
This theorem is referenced by:  2reu7  40492  2reu8  40493
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