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Theorem 2uasban 44582
Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2uasban (∃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)

Proof of Theorem 2uasban
StepHypRef Expression
1 biid 261 . 2 ((∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
212uasbanh 44581 1 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482
This theorem is referenced by: (None)
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