Theorem pm14.123c 44459

Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Assertion

Ref	Expression
pm14.123c	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))

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

Step	Hyp	Ref	Expression
1		pm14.123a 44457	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
2		pm14.123b 44458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))
3	1, 2	bitrd 279	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  [wsbc 3741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sbc 3742

This theorem is referenced by: (None)