Theorem pm11.71 41904

Description: Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.71	⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem pm11.71

Step	Hyp	Ref	Expression
1		nfv 1918	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓)
2		nfv 1918	. . . 4 ⊢ Ⅎ𝑥(𝜒 → 𝜃)
3	1, 2	aaan 2332	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)))
4		anim12 805	. . . 4 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
5	4	2alimi 1816	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
6	3, 5	sylbir 234	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) → ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
7		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥𝜒
8	7	nfex 2322	. . . . 5 ⊢ Ⅎ𝑥∃𝑦𝜒
9		exim 1837	. . . . . . 7 ⊢ (∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → (∃𝑦(𝜑 ∧ 𝜒) → ∃𝑦(𝜓 ∧ 𝜃)))
10		19.42v 1958	. . . . . . 7 ⊢ (∃𝑦(𝜑 ∧ 𝜒) ↔ (𝜑 ∧ ∃𝑦𝜒))
11		19.42v 1958	. . . . . . 7 ⊢ (∃𝑦(𝜓 ∧ 𝜃) ↔ (𝜓 ∧ ∃𝑦𝜃))
12	9, 10, 11	3imtr3g 294	. . . . . 6 ⊢ (∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ((𝜑 ∧ ∃𝑦𝜒) → (𝜓 ∧ ∃𝑦𝜃)))
13		pm3.21 471	. . . . . . 7 ⊢ (∃𝑦𝜒 → (𝜑 → (𝜑 ∧ ∃𝑦𝜒)))
14		simpl 482	. . . . . . . 8 ⊢ ((𝜓 ∧ ∃𝑦𝜃) → 𝜓)
15	14	imim2i 16	. . . . . . 7 ⊢ (((𝜑 ∧ ∃𝑦𝜒) → (𝜓 ∧ ∃𝑦𝜃)) → ((𝜑 ∧ ∃𝑦𝜒) → 𝜓))
16	13, 15	syl9 77	. . . . . 6 ⊢ (∃𝑦𝜒 → (((𝜑 ∧ ∃𝑦𝜒) → (𝜓 ∧ ∃𝑦𝜃)) → (𝜑 → 𝜓)))
17	12, 16	syl5 34	. . . . 5 ⊢ (∃𝑦𝜒 → (∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → (𝜑 → 𝜓)))
18	8, 17	alimd 2208	. . . 4 ⊢ (∃𝑦𝜒 → (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑥(𝜑 → 𝜓)))
19	18	adantl 481	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑥(𝜑 → 𝜓)))
20		ax-11 2156	. . . . 5 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑦∀𝑥((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
21		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
22	21	nfex 2322	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥𝜑
23		exim 1837	. . . . . . . 8 ⊢ (∀𝑥((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜓 ∧ 𝜃)))
24		19.41v 1954	. . . . . . . 8 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ 𝜒))
25		19.41v 1954	. . . . . . . 8 ⊢ (∃𝑥(𝜓 ∧ 𝜃) ↔ (∃𝑥𝜓 ∧ 𝜃))
26	23, 24, 25	3imtr3g 294	. . . . . . 7 ⊢ (∀𝑥((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ((∃𝑥𝜑 ∧ 𝜒) → (∃𝑥𝜓 ∧ 𝜃)))
27		pm3.2 469	. . . . . . . 8 ⊢ (∃𝑥𝜑 → (𝜒 → (∃𝑥𝜑 ∧ 𝜒)))
28		simpr 484	. . . . . . . . 9 ⊢ ((∃𝑥𝜓 ∧ 𝜃) → 𝜃)
29	28	imim2i 16	. . . . . . . 8 ⊢ (((∃𝑥𝜑 ∧ 𝜒) → (∃𝑥𝜓 ∧ 𝜃)) → ((∃𝑥𝜑 ∧ 𝜒) → 𝜃))
30	27, 29	syl9 77	. . . . . . 7 ⊢ (∃𝑥𝜑 → (((∃𝑥𝜑 ∧ 𝜒) → (∃𝑥𝜓 ∧ 𝜃)) → (𝜒 → 𝜃)))
31	26, 30	syl5 34	. . . . . 6 ⊢ (∃𝑥𝜑 → (∀𝑥((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → (𝜒 → 𝜃)))
32	22, 31	alimd 2208	. . . . 5 ⊢ (∃𝑥𝜑 → (∀𝑦∀𝑥((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑦(𝜒 → 𝜃)))
33	20, 32	syl5 34	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑦(𝜒 → 𝜃)))
34	33	adantr 480	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → ∀𝑦(𝜒 → 𝜃)))
35	19, 34	jcad 512	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → (∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)) → (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃))))
36	6, 35	impbid2 225	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)