Theorem sn-axrep5v 42255

Description: A condensed form of axrep5 5287. (Contributed by SN, 21-Sep-2023.)

Assertion

Ref	Expression
sn-axrep5v	⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤)

Proof of Theorem sn-axrep5v

Step	Hyp	Ref	Expression
1		axrep6 5288	. 2 ⊢ (∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ 𝜑) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝑤 ∈ 𝑥 ∧ 𝜑)))
2		19.37v 1991	. . . . 5 ⊢ (∃𝑦(𝑤 ∈ 𝑥 → ∀𝑧(𝜑 → 𝑧 = 𝑦)) ↔ (𝑤 ∈ 𝑥 → ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦)))
3		impexp 450	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ (𝑤 ∈ 𝑥 → (𝜑 → 𝑧 = 𝑦)))
4	3	albii 1819	. . . . . . 7 ⊢ (∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤 ∈ 𝑥 → (𝜑 → 𝑧 = 𝑦)))
5		19.21v 1939	. . . . . . 7 ⊢ (∀𝑧(𝑤 ∈ 𝑥 → (𝜑 → 𝑧 = 𝑦)) ↔ (𝑤 ∈ 𝑥 → ∀𝑧(𝜑 → 𝑧 = 𝑦)))
6	4, 5	bitri 275	. . . . . 6 ⊢ (∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ (𝑤 ∈ 𝑥 → ∀𝑧(𝜑 → 𝑧 = 𝑦)))
7	6	exbii 1848	. . . . 5 ⊢ (∃𝑦∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ ∃𝑦(𝑤 ∈ 𝑥 → ∀𝑧(𝜑 → 𝑧 = 𝑦)))
8		df-mo 2540	. . . . . 6 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
9	8	imbi2i 336	. . . . 5 ⊢ ((𝑤 ∈ 𝑥 → ∃*𝑧𝜑) ↔ (𝑤 ∈ 𝑥 → ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦)))
10	2, 7, 9	3bitr4i 303	. . . 4 ⊢ (∃𝑦∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ (𝑤 ∈ 𝑥 → ∃*𝑧𝜑))
11	10	albii 1819	. . 3 ⊢ (∀𝑤∃𝑦∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦) ↔ ∀𝑤(𝑤 ∈ 𝑥 → ∃*𝑧𝜑))
12		df-mo 2540	. . . 4 ⊢ (∃*𝑧(𝑤 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑦∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦))
13	12	albii 1819	. . 3 ⊢ (∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ 𝜑) ↔ ∀𝑤∃𝑦∀𝑧((𝑤 ∈ 𝑥 ∧ 𝜑) → 𝑧 = 𝑦))
14		df-ral 3062	. . 3 ⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 ↔ ∀𝑤(𝑤 ∈ 𝑥 → ∃*𝑧𝜑))
15	11, 13, 14	3bitr4i 303	. 2 ⊢ (∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ 𝜑) ↔ ∀𝑤 ∈ 𝑥 ∃*𝑧𝜑)
16		rexanid 3096	. . . . 5 ⊢ (∃𝑤 ∈ 𝑥 (𝑤 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑤 ∈ 𝑥 𝜑)
17	16	bibi2i 337	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝑤 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
18	17	albii 1819	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝑤 ∈ 𝑥 ∧ 𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
19	18	exbii 1848	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝑤 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
20	1, 15, 19	3imtr3i 291	1 ⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  ∃*wmo 2538  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-rep 5279

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2540  df-ral 3062  df-rex 3071

This theorem is referenced by: (None)