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Theorem pm14.123a 40634
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 1882 . 2 (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
2 2sbc6g 40624 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
32anbi2d 628 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
41, 3syl5bb 284 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 207  wa 396  wal 1526   = wceq 1528  wcel 2105  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  pm14.123c  40636
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