Theorem sscoll2 10113

Description: Version of ax-sscoll 10112 with two DV 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 ∀𝑦(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		nfv 1421	. . 3 ⊢ Ⅎ𝑐(𝑢 = 𝑎 ∧ 𝑣 = 𝑏)
2		nfv 1421	. . . 4 ⊢ Ⅎ𝑧(𝑢 = 𝑎 ∧ 𝑣 = 𝑏)
3		simpl 102	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑢 = 𝑎)
4		rexeq 2506	. . . . . . 7 ⊢ (𝑣 = 𝑏 → (∃𝑦 ∈ 𝑣 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
5	4	adantl 262	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑦 ∈ 𝑣 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
6	3, 5	raleqbidv 2517	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
7		nfv 1421	. . . . . 6 ⊢ Ⅎ𝑑(𝑢 = 𝑎 ∧ 𝑣 = 𝑏)
8		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑦(𝑢 = 𝑎 ∧ 𝑣 = 𝑏)
9		rexeq 2506	. . . . . . . . 9 ⊢ (𝑢 = 𝑎 → (∃𝑥 ∈ 𝑢 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
10	9	adantr 261	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑥 ∈ 𝑢 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
11	10	bibi2d 221	. . . . . . 7 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑) ↔ (𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
12	8, 11	albid 1506	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
13	7, 12	rexbid 2325	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑) ↔ ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
14	6, 13	imbi12d 223	. . . 4 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))))
15	2, 14	albid 1506	. . 3 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑)) ↔ ∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))))
16	1, 15	exbid 1507	. 2 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑)) ↔ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))))
17		ax-sscoll 10112	. . . 4 ⊢ ∀𝑢∀𝑣∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑))
18	17	spi 1429	. . 3 ⊢ ∀𝑣∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑))
19	18	spi 1429	. 2 ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝑣 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑢 𝜑))
20	16, 19	ch2varv 9908	1 ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sscoll 10112

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312

This theorem is referenced by: (None)