Theorem pm14.123a 42271

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

Assertion

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

Step	Hyp	Ref	Expression
1		2albiim 1892	. 2 ⊢ (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
2		2sbc6g 42261	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
3	2	anbi2d 629	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
4	1, 3	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538   = wceq 1540   ∈ wcel 2105  [wsbc 3725

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-sbc 3726

This theorem is referenced by:  pm14.123c  42273