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Theorem bj-axrep3 33620
 Description: Remove dependency on ax-13 2301 from axrep3 5054. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axrep3 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem bj-axrep3
StepHypRef Expression
1 nfe1 2087 . . . 4 𝑦𝑦𝑧(𝜑𝑧 = 𝑦)
2 nfv 1873 . . . . . 6 𝑦 𝑧𝑥
3 nfv 1873 . . . . . . . 8 𝑦 𝑥𝑤
4 nfa1 2088 . . . . . . . 8 𝑦𝑦𝜑
53, 4nfan 1862 . . . . . . 7 𝑦(𝑥𝑤 ∧ ∀𝑦𝜑)
65nfex 2264 . . . . . 6 𝑦𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
72, 6nfbi 1866 . . . . 5 𝑦(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
87nfal 2263 . . . 4 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
91, 8nfim 1859 . . 3 𝑦(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
109nfex 2264 . 2 𝑦𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
11 elequ2 2064 . . . . . . . 8 (𝑦 = 𝑤 → (𝑥𝑦𝑥𝑤))
1211anbi1d 620 . . . . . . 7 (𝑦 = 𝑤 → ((𝑥𝑦 ∧ ∀𝑦𝜑) ↔ (𝑥𝑤 ∧ ∀𝑦𝜑)))
1312exbidv 1880 . . . . . 6 (𝑦 = 𝑤 → (∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
1413bibi2d 335 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))))
1514albidv 1879 . . . 4 (𝑦 = 𝑤 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))))
1615imbi2d 333 . . 3 (𝑦 = 𝑤 → ((∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
1716exbidv 1880 . 2 (𝑦 = 𝑤 → (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
18 bj-axrep2 33619 . 2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
1910, 17, 18bj-chvarv 33573 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-rep 5050 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747 This theorem is referenced by:  bj-axrep4  33621
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