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Theorem 2eu3OLD 2685
 Description: Obsolete version of 2eu3 2684 as of 23-Apr-2023. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2eu3OLD (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu3OLD
StepHypRef Expression
1 nfmo1 2570 . . . . 5 𝑦∃*𝑦𝜑
2119.31 2166 . . . 4 (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
32albii 1782 . . 3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4 nfmo1 2570 . . . . 5 𝑥∃*𝑥𝜑
54nfal 2263 . . . 4 𝑥𝑦∃*𝑥𝜑
6519.32 2165 . . 3 (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
73, 6bitri 267 . 2 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8 2eu1 2681 . . . . . . 7 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
98biimpd 221 . . . . . 6 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
10 ancom 453 . . . . . 6 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
119, 10syl6ib 243 . . . . 5 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1211adantld 483 . . . 4 (∀𝑦∃*𝑥𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
13 2eu1 2681 . . . . . 6 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1413biimpd 221 . . . . 5 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1514adantrd 484 . . . 4 (∀𝑥∃*𝑦𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1612, 15jaoi 843 . . 3 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
17 2exeu 2677 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
18 2exeu 2677 . . . . 5 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
1918ancoms 451 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
2017, 19jca 504 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
2116, 20impbid1 217 . 2 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
227, 21sylbi 209 1 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833  ∀wal 1505  ∃wex 1742  ∃*wmo 2545  ∃!weu 2583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584 This theorem is referenced by: (None)
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