Theorem sscoll2 14743

Description: Version of ax-sscoll 14742 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)

Assertion

Ref	Expression
sscoll2	⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑥,𝑦,𝑧   𝜑,𝑐,𝑑

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem sscoll2

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑢 = 𝑎)
2		rexeq 2674	. . . . . . 7 ⊢ (𝑣 = 𝑏 → (∃𝑦 ∈ 𝑣 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
3	2	adantl 277	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑦 ∈ 𝑣 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
4	1, 3	raleqbidv 2685	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
5		eleq2 2241	. . . . . . . . . 10 ⊢ (𝑢 = 𝑎 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑎))
6	5	adantr 276	. . . . . . . . 9 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑎))
7	6	imbi1d 231	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((𝑥 ∈ 𝑢 → ∃𝑦 ∈ 𝑑 𝜑) ↔ (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑑 𝜑)))
8	7	ralbidv2 2479	. . . . . . 7 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑))
9	6	anbi1d 465	. . . . . . . . 9 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((𝑥 ∈ 𝑢 ∧ 𝜑) ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
10	9	rexbidv2 2480	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑥 ∈ 𝑢 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
11	10	ralbidv 2477	. . . . . . 7 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑 ↔ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))
12	8, 11	anbi12d 473	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)))
13	12	rexbidv 2478	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑) ↔ ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)))
14	4, 13	imbi12d 234	. . . 4 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))))
15	14	albidv 1824	. . 3 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑)) ↔ ∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))))
16	15	exbidv 1825	. 2 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑)) ↔ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))))
17		ax-sscoll 14742	. . . 4 ⊢ ∀𝑢∀𝑣∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑))
18	17	spi 1536	. . 3 ⊢ ∀𝑣∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑))
19	18	spi 1536	. 2 ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑))
20	16, 19	ch2varv 14523	1 ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  ∃wex 1492  ∀wral 2455  ∃wrex 2456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sscoll 14742

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461

This theorem is referenced by: (None)