Theorem pm14.123b 45002

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

Assertion

Ref	Expression
pm14.123b	⊢ ((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)))

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

Step	Hyp	Ref	Expression
1		2sbc5g 44992	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
2	1	adantr 484	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵))) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
3		nfa1 2185	. . . . 5 ⊢ Ⅎ𝑧∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵))
4		nfa2 2209	. . . . . 6 ⊢ Ⅎ𝑤∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵))
5		simpr 488	. . . . . . 7 ⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) → 𝜑)
6		2sp 2221	. . . . . . . 8 ⊢ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) → (𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
7	6	ancrd 559	. . . . . . 7 ⊢ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) → (𝜑 → ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
8	5, 7	impbid2 228	. . . . . 6 ⊢ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) → (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ 𝜑))
9	4, 8	exbid 2258	. . . . 5 ⊢ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) → (∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑤𝜑))
10	3, 9	exbid 2258	. . . 4 ⊢ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧∃𝑤𝜑))
11	10	adantl 485	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵))) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧∃𝑤𝜑))
12	2, 11	bitr3d 283	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵))) → ([𝐴 / 𝑧][𝐵 / 𝑤]𝜑 ↔ ∃𝑧∃𝑤𝜑))
13	12	pm5.32da 587	1 ⊢ ((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)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-sbc 3745

This theorem is referenced by:  pm14.123c  45003