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Theorem pm14.123c 40749
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123c ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∃𝑧𝑤𝜑)))
Distinct variable groups:   𝑤,𝐴,𝑧   𝑤,𝐵,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑉(𝑧,𝑤)   𝑊(𝑧,𝑤)

Proof of Theorem pm14.123c
StepHypRef Expression
1 pm14.123a 40747 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
2 pm14.123b 40748 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∃𝑧𝑤𝜑)))
31, 2bitrd 281 1 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∃𝑧𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wex 1774  wcel 2108  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-v 3495  df-sbc 3771
This theorem is referenced by: (None)
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