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Theorem pm14.123b 43175
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 43165 . . . 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 2148 . . . . 5 𝑧𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
4 nfa2 2170 . . . . . 6 𝑤𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
5 simpr 485 . . . . . . 7 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) → 𝜑)
6 2sp 2179 . . . . . . . 8 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)))
76ancrd 552 . . . . . . 7 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → ((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑)))
85, 7impbid2 225 . . . . . 6 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ 𝜑))
94, 8exbid 2216 . . . . 5 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (∃𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑤𝜑))
103, 9exbid 2216 . . . 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 280 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → ([𝐴 / 𝑧][𝐵 / 𝑤]𝜑 ↔ ∃𝑧𝑤𝜑))
1312pm5.32da 579 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 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sbc 3778
This theorem is referenced by:  pm14.123c  43176
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