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Theorem pm14.123b 38144
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123b ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀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 pm14.123b
StepHypRef Expression
1 2sbc5g 38134 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
21adantr 481 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
3 nfa1 2025 . . . . 5 𝑧𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
4 nfa2 2037 . . . . . 6 𝑤𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
5 simpr 477 . . . . . . 7 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) → 𝜑)
6 2sp 2054 . . . . . . . 8 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)))
76ancrd 576 . . . . . . 7 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → ((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑)))
85, 7impbid2 216 . . . . . 6 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ 𝜑))
94, 8exbid 2089 . . . . 5 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (∃𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑤𝜑))
103, 9exbid 2089 . . . 4 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧𝑤𝜑))
1110adantl 482 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧𝑤𝜑))
122, 11bitr3d 270 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → ([𝐴 / 𝑧][𝐵 / 𝑤]𝜑 ↔ ∃𝑧𝑤𝜑))
1312pm5.32da 672 1 ((𝐴𝑉𝐵𝑊) → ((∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) ∧ ∃𝑧𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422
This theorem is referenced by:  pm14.123c  38145
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