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Theorem 2uasbanh 38603
Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 38603 is derived from 2uasbanhVD 38973. (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 473 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → (𝑥 = 𝑢𝑦 = 𝑣))
2 simprl 794 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → 𝜑)
31, 2jca 554 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
432eximi 1762 . . 3 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
5 simprr 796 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → 𝜓)
61, 5jca 554 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
762eximi 1762 . . 3 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
84, 7jca 554 . 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 476 . . . . . 6 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
11 simpl 473 . . . . . . . . . 10 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
12112eximi 1762 . . . . . . . . 9 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
1310, 12syl 17 . . . . . . . 8 (𝜒 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
14 ax6e2ndeq 38601 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
1513, 14sylibr 224 . . . . . . 7 (𝜒 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
16 2sb5nd 38602 . . . . . . 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 247 . . . . 5 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
199simprbi 480 . . . . . 6 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓))
20 2sb5nd 38602 . . . . . . 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 247 . . . . 5 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜓)
23 sban 2398 . . . . . . 7 ([𝑣 / 𝑦](𝜑𝜓) ↔ ([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
2423sbbii 1886 . . . . . 6 ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
25 sban 2398 . . . . . 6 ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
2624, 25bitri 264 . . . . 5 ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
2718, 22, 26sylanbrc 698 . . . 4 (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦](𝜑𝜓))
28 2sb5nd 38602 . . . . 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 222 . . 3 (𝜒 → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)))
319, 30sylbir 225 . 2 ((∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)) → ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)))
328, 31impbii 199 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 196  wo 383  wa 384  wal 1480  wex 1703  [wsb 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2794  df-v 3200
This theorem is referenced by:  2uasban  38604
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