Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2uasbanh Structured version   Visualization version   GIF version

Theorem 2uasbanh 41640
 Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 41640 is derived from 2uasbanhVD 41990. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
2uasbanh.1 (𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
Assertion
Ref Expression
2uasbanh (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem 2uasbanh
StepHypRef Expression
1 simpl 486 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → (𝑥 = 𝑢𝑦 = 𝑣))
2 simprl 770 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → 𝜑)
31, 2jca 515 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
432eximi 1837 . . 3 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
5 simprr 772 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → 𝜓)
61, 5jca 515 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
762eximi 1837 . . 3 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
84, 7jca 515 . 2 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
9 2uasbanh.1 . . 3 (𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
109simplbi 501 . . . . . 6 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
11 simpl 486 . . . . . . . . . 10 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
12112eximi 1837 . . . . . . . . 9 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
1310, 12syl 17 . . . . . . . 8 (𝜒 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
14 ax6e2ndeq 41638 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
1513, 14sylibr 237 . . . . . . 7 (𝜒 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
16 2sb5nd 41639 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
1715, 16syl 17 . . . . . 6 (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
1810, 17mpbird 260 . . . . 5 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
199simprbi 500 . . . . . 6 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
20 2sb5nd 41639 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
2115, 20syl 17 . . . . . 6 (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
2219, 21mpbird 260 . . . . 5 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜓)
23 sban 2085 . . . . . . 7 ([𝑣 / 𝑦](𝜑𝜓) ↔ ([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
2423sbbii 2081 . . . . . 6 ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
25 sban 2085 . . . . . 6 ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
2624, 25bitri 278 . . . . 5 ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
2718, 22, 26sylanbrc 586 . . . 4 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓))
28 2sb5nd 41639 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓))))
2915, 28syl 17 . . . 4 (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓))))
3027, 29mpbid 235 . . 3 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)))
319, 30sylbir 238 . 2 ((∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)))
328, 31impbii 212 1 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536  ∃wex 1781  [wsb 2069 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-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411 This theorem is referenced by:  2uasban  41641
 Copyright terms: Public domain W3C validator