Theorem axrep6 5294

Description: A condensed form of ax-rep 5285. (Contributed by SN, 18-Sep-2023.) (Proof shortened by Matthew House, 18-Sep-2025.)

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		axrep4v 5290	. 2 ⊢ (∀𝑤∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝜑)))
2		df-mo 2538	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
3	2	albii 1816	. 2 ⊢ (∀𝑤∃*𝑧𝜑 ↔ ∀𝑤∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦))
4		df-rex 3069	. . . . 5 ⊢ (∃𝑤 ∈ 𝑥 𝜑 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝜑))
5	4	bibi2i 337	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝜑)))
6	5	albii 1816	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝜑)))
7	6	exbii 1845	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝜑)))
8	1, 3, 7	3imtr4i 292	1 ⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1776  ∃*wmo 2536  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-rep 5285

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-rex 3069

This theorem is referenced by:  axrep6g  5296  axsepgfromrep  5300  sn-axrep5v  42234  sn-axprlem3  42235