Theorem axrep6 5212

Description: A condensed form of ax-rep 5205. (Contributed by SN, 18-Sep-2023.)

Assertion

Ref	Expression
axrep6	⊢ (∀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 axrep6

Step	Hyp	Ref	Expression
1		ax-rep 5205	. 2 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
2		df-mo 2540	. . . 4 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
3		19.3v 1986	. . . . . . 7 ⊢ (∀𝑦𝜑 ↔ 𝜑)
4	3	imbi1i 349	. . . . . 6 ⊢ ((∀𝑦𝜑 → 𝑧 = 𝑦) ↔ (𝜑 → 𝑧 = 𝑦))
5	4	albii 1823	. . . . 5 ⊢ (∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∀𝑧(𝜑 → 𝑧 = 𝑦))
6	5	exbii 1851	. . . 4 ⊢ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
7	2, 6	bitr4i 277	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
8	7	albii 1823	. 2 ⊢ (∀𝑤∃*𝑧𝜑 ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦))
9	3	rexbii 3177	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑥 ∀𝑦𝜑 ↔ ∃𝑤 ∈ 𝑥 𝜑)
10		df-rex 3069	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑥 ∀𝑦𝜑 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
11	9, 10	bitr3i 276	. . . . 5 ⊢ (∃𝑤 ∈ 𝑥 𝜑 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
12	11	bibi2i 337	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
13	12	albii 1823	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
14	13	exbii 1851	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
15	1, 8, 14	3imtr4i 291	1 ⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  ∃*wmo 2538  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-rep 5205

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-rex 3069

This theorem is referenced by:  axsepgfromrep  5216  sn-axrep5v  40113  sn-axprlem3  40114