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Theorem pm14.123a 39403
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123a ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
Distinct variable groups:   𝑤,𝐴,𝑧   𝑤,𝐵,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑉(𝑧,𝑤)   𝑊(𝑧,𝑤)

Proof of Theorem pm14.123a
StepHypRef Expression
1 2albiim 1989 . 2 (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
2 2sbc6g 39393 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
32anbi2d 623 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
41, 3syl5bb 275 1 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634
This theorem is referenced by:  pm14.123c  39405
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