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Theorem sn-axrep5v 39387
Description: A condensed form of axrep5 5163. (Contributed by SN, 21-Sep-2023.)
Assertion
Ref Expression
sn-axrep5v (∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤)

Proof of Theorem sn-axrep5v
StepHypRef Expression
1 axrep6 5164 . 2 (∀𝑤∃*𝑧(𝑤𝑥𝜑) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 (𝑤𝑥𝜑)))
2 19.37v 1998 . . . . 5 (∃𝑦(𝑤𝑥 → ∀𝑧(𝜑𝑧 = 𝑦)) ↔ (𝑤𝑥 → ∃𝑦𝑧(𝜑𝑧 = 𝑦)))
3 impexp 454 . . . . . . . 8 (((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ (𝑤𝑥 → (𝜑𝑧 = 𝑦)))
43albii 1821 . . . . . . 7 (∀𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝑥 → (𝜑𝑧 = 𝑦)))
5 19.21v 1940 . . . . . . 7 (∀𝑧(𝑤𝑥 → (𝜑𝑧 = 𝑦)) ↔ (𝑤𝑥 → ∀𝑧(𝜑𝑧 = 𝑦)))
64, 5bitri 278 . . . . . 6 (∀𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ (𝑤𝑥 → ∀𝑧(𝜑𝑧 = 𝑦)))
76exbii 1849 . . . . 5 (∃𝑦𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ ∃𝑦(𝑤𝑥 → ∀𝑧(𝜑𝑧 = 𝑦)))
8 df-mo 2601 . . . . . 6 (∃*𝑧𝜑 ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦))
98imbi2i 339 . . . . 5 ((𝑤𝑥 → ∃*𝑧𝜑) ↔ (𝑤𝑥 → ∃𝑦𝑧(𝜑𝑧 = 𝑦)))
102, 7, 93bitr4i 306 . . . 4 (∃𝑦𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ (𝑤𝑥 → ∃*𝑧𝜑))
1110albii 1821 . . 3 (∀𝑤𝑦𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦) ↔ ∀𝑤(𝑤𝑥 → ∃*𝑧𝜑))
12 df-mo 2601 . . . 4 (∃*𝑧(𝑤𝑥𝜑) ↔ ∃𝑦𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦))
1312albii 1821 . . 3 (∀𝑤∃*𝑧(𝑤𝑥𝜑) ↔ ∀𝑤𝑦𝑧((𝑤𝑥𝜑) → 𝑧 = 𝑦))
14 df-ral 3114 . . 3 (∀𝑤𝑥 ∃*𝑧𝜑 ↔ ∀𝑤(𝑤𝑥 → ∃*𝑧𝜑))
1511, 13, 143bitr4i 306 . 2 (∀𝑤∃*𝑧(𝑤𝑥𝜑) ↔ ∀𝑤𝑥 ∃*𝑧𝜑)
16 rexanid 3218 . . . . 5 (∃𝑤𝑥 (𝑤𝑥𝜑) ↔ ∃𝑤𝑥 𝜑)
1716bibi2i 341 . . . 4 ((𝑧𝑦 ↔ ∃𝑤𝑥 (𝑤𝑥𝜑)) ↔ (𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
1817albii 1821 . . 3 (∀𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 (𝑤𝑥𝜑)) ↔ ∀𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
1918exbii 1849 . 2 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 (𝑤𝑥𝜑)) ↔ ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
201, 15, 193imtr3i 294 1 (∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  ∃*wmo 2599  wral 3109  wrex 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-rep 5157
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-ral 3114  df-rex 3115
This theorem is referenced by: (None)
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