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Theorem bj-axrep2 33117
Description: Remove dependency on ax-13 2391 from axrep2 4925. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axrep2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem bj-axrep2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfe1 2176 . . . . 5 𝑤𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤)
2 nfv 1992 . . . . 5 𝑤𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))
31, 2nfim 1974 . . . 4 𝑤(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
43nfex 2301 . . 3 𝑤𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
5 elequ2 2153 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
65anbi1d 743 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑥𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
76exbidv 1999 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
87bibi2d 331 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
98albidv 1998 . . . . 5 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
109imbi2d 329 . . . 4 (𝑤 = 𝑦 → ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ (∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
1110exbidv 1999 . . 3 (𝑤 = 𝑦 → (∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
12 bj-axrep1 33116 . . 3 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
134, 11, 12bj-chvarv 33053 . 2 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
14 sp 2200 . . . . . . 7 (∀𝑦𝜑𝜑)
1514imim1i 63 . . . . . 6 ((𝜑𝑧 = 𝑦) → (∀𝑦𝜑𝑧 = 𝑦))
1615alimi 1888 . . . . 5 (∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑𝑧 = 𝑦))
1716eximi 1911 . . . 4 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦))
18 nfv 1992 . . . . 5 𝑤𝑧(∀𝑦𝜑𝑧 = 𝑦)
19 nfa1 2177 . . . . . . 7 𝑦𝑦𝜑
20 nfv 1992 . . . . . . 7 𝑦 𝑧 = 𝑤
2119, 20nfim 1974 . . . . . 6 𝑦(∀𝑦𝜑𝑧 = 𝑤)
2221nfal 2300 . . . . 5 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑤)
23 equequ2 2108 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 = 𝑦𝑧 = 𝑤))
2423imbi2d 329 . . . . . 6 (𝑦 = 𝑤 → ((∀𝑦𝜑𝑧 = 𝑦) ↔ (∀𝑦𝜑𝑧 = 𝑤)))
2524albidv 1998 . . . . 5 (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑𝑧 = 𝑤)))
2618, 22, 25cbvexv1 2321 . . . 4 (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2717, 26sylib 208 . . 3 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2827imim1i 63 . 2 ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
2913, 28eximii 1913 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-rep 4923
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859
This theorem is referenced by:  bj-axrep3  33118
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