Theorem axrep6OLD 5250

Description: Obsolete version of axrep6 5249 as of 18-Sep-2025. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axrep6OLD	⊢ (∀𝑤∃*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 axrep6OLD

Step	Hyp	Ref	Expression
1		ax-rep 5240	. 2 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
2		dfmo 2570	. . . 4 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
3		19.3v 2015	. . . . . . 7 ⊢ (∀𝑦𝜑 ↔ 𝜑)
4	3	imbi1i 352	. . . . . 6 ⊢ ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (𝜑 → 𝑧 = 𝑦))
5	4	albii 1852	. . . . 5 ⊢ (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(𝜑 → 𝑧 = 𝑦))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
7	2, 6	bitr4i 281	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
8	7	albii 1852	. 2 ⊢ (∀𝑤∃*𝑧𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
9	3	rexbii 3114	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑥 ∀𝑦𝜑 ↔ ∃𝑤 ∈ 𝑥 𝜑)
10		df-rex 3092	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑥 ∀𝑦𝜑 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
11	9, 10	bitr3i 280	. . . . 5 ⊢ (∃𝑤 ∈ 𝑥 𝜑 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
12	11	bibi2i 340	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
13	12	albii 1852	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
14	13	exbii 1881	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
15	1, 8, 14	3imtr4i 295	1 ⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∃*wmo 2567  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-rep 5240

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-rex 3092

This theorem is used by: (None)