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Theorem pm14.123a 37446
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 1805 . 2 (∀𝑧𝑤(𝜑 ↔ (𝑧 = 𝐴𝑤 = 𝐵)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
2 2sbc6g 37436 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
32anbi2d 735 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
41, 3syl5bb 270 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 194  wa 382  wal 1472   = wceq 1474  wcel 1975  [wsbc 3396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-sbc 3397
This theorem is referenced by:  pm14.123c  37448
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