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Theorem 2reu2 41693
Description: Double restricted existential uniqueness, analogous to 2eu2 2692. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Assertion
Ref Expression
2reu2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem 2reu2
StepHypRef Expression
1 reurmo 3300 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑)
2 2rmorex 3553 . . 3 (∃*𝑦𝐵𝑥𝐴 𝜑 → ∀𝑥𝐴 ∃*𝑦𝐵 𝜑)
3 2reu1 41692 . . . 4 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))
4 simpl 474 . . . 4 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴𝑦𝐵 𝜑)
53, 4syl6bi 243 . . 3 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
7 2rexreu 41691 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
87expcom 450 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑))
96, 8impbid 202 1 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wral 3050  wrex 3051  ∃!wreu 3052  ∃*wrmo 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058
This theorem is referenced by:  2reu8  41698
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