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Theorem pm14.123b 41173
 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 41163 . . . 4 ((𝐴𝑉𝐵𝑊) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
21adantr 484 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
3 nfa1 2152 . . . . 5 𝑧𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
4 nfa2 2174 . . . . . 6 𝑤𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))
5 simpr 488 . . . . . . 7 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) → 𝜑)
6 2sp 2183 . . . . . . . 8 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)))
76ancrd 555 . . . . . . 7 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (𝜑 → ((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑)))
85, 7impbid2 229 . . . . . 6 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ 𝜑))
94, 8exbid 2223 . . . . 5 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (∃𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑤𝜑))
103, 9exbid 2223 . . . 4 (∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵)) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧𝑤𝜑))
1110adantl 485 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧𝑤𝜑))
122, 11bitr3d 284 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∀𝑧𝑤(𝜑 → (𝑧 = 𝐴𝑤 = 𝐵))) → ([𝐴 / 𝑧][𝐵 / 𝑤]𝜑 ↔ ∃𝑧𝑤𝜑))
1312pm5.32da 582 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 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  [wsbc 3720 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721 This theorem is referenced by:  pm14.123c  41174
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