Theorem sscoll2 15480

Description: Version of ax-sscoll 15479 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 2691	. . . . . . 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 2706	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
5		eleq2 2257	. . . . . . . . . 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 2496	. . . . . . 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 2497	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑥 ∈ 𝑢 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
11	10	ralbidv 2494	. . . . . . 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 2495	. . . . 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 1835	. . 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 1836	. 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 15479	. . . 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 1547	. . 3 ⊢ ∀𝑣∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑))
19	18	spi 1547	. 2 ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑢 𝜑))
20	16, 19	ch2varv 15260	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 1362  ∃wex 1503  ∀wral 2472  ∃wrex 2473

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sscoll 15479

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478

This theorem is referenced by: (None)