Theorem 2uasbanh 42070

Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 42070 is derived from 2uasbanhVD 42420. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
2uasbanh.1	⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))

Assertion

Ref	Expression
2uasbanh	⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem 2uasbanh

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
2		simprl 767	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → 𝜑)
3	1, 2	jca 511	. . . 4 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
4	3	2eximi 1839	. . 3 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
5		simprr 769	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → 𝜓)
6	1, 5	jca 511	. . . 4 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
7	6	2eximi 1839	. . 3 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
8	4, 7	jca 511	. 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
9		2uasbanh.1	. . 3 ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
10	9	simplbi 497	. . . . . 6 ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
11		simpl 482	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
12	11	2eximi 1839	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
13	10, 12	syl 17	. . . . . . . 8 ⊢ (𝜒 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
14		ax6e2ndeq 42068	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
15	13, 14	sylibr 233	. . . . . . 7 ⊢ (𝜒 → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
16		2sb5nd 42069	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
18	10, 17	mpbird 256	. . . . 5 ⊢ (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
19	9	simprbi 496	. . . . . 6 ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
20		2sb5nd 42069	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
21	15, 20	syl 17	. . . . . 6 ⊢ (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
22	19, 21	mpbird 256	. . . . 5 ⊢ (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜓)
23		sban 2084	. . . . . . 7 ⊢ ([𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
24	23	sbbii 2080	. . . . . 6 ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
25		sban 2084	. . . . . 6 ⊢ ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
26	24, 25	bitri 274	. . . . 5 ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
27	18, 22, 26	sylanbrc 582	. . . 4 ⊢ (𝜒 → [𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓))
28		2sb5nd 42069	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))))
29	15, 28	syl 17	. . . 4 ⊢ (𝜒 → ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))))
30	27, 29	mpbid 231	. . 3 ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))
31	9, 30	sylbir 234	. 2 ⊢ ((∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))
32	8, 31	impbii 208	1 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843  ∀wal 1537  ∃wex 1783  [wsb 2068

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-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424

This theorem is referenced by:  2uasban  42071