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Theorem bj-axrep1 33624
 Description: Remove dependency on ax-13 2301 from axrep1 5050. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axrep1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem bj-axrep1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2064 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
21anbi1d 620 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑥𝑤𝜑) ↔ (𝑥𝑦𝜑)))
32exbidv 1880 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑥𝑤𝜑) ↔ ∃𝑥(𝑥𝑦𝜑)))
43bibi2d 335 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
54albidv 1879 . . . . 5 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
65exbidv 1880 . . . 4 (𝑤 = 𝑦 → (∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
76imbi2d 333 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))) ↔ (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))))
8 ax-rep 5049 . . . 4 (∀𝑥𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
9 19.3v 1939 . . . . . . . 8 (∀𝑦𝜑𝜑)
109imbi1i 342 . . . . . . 7 ((∀𝑦𝜑𝑧 = 𝑦) ↔ (𝜑𝑧 = 𝑦))
1110albii 1782 . . . . . 6 (∀𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑧(𝜑𝑧 = 𝑦))
1211exbii 1810 . . . . 5 (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦))
1312albii 1782 . . . 4 (∀𝑥𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦))
14 nfv 1873 . . . . . . 7 𝑥 𝑧𝑦
15 nfe1 2087 . . . . . . 7 𝑥𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
1614, 15nfbi 1866 . . . . . 6 𝑥(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
1716nfal 2263 . . . . 5 𝑥𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
18 nfv 1873 . . . . 5 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))
19 elequ2 2064 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
209anbi2i 613 . . . . . . . . 9 ((𝑥𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥𝑤𝜑))
2120exbii 1810 . . . . . . . 8 (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
2221a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2319, 22bibi12d 338 . . . . . 6 (𝑦 = 𝑥 → ((𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))))
2423albidv 1879 . . . . 5 (𝑦 = 𝑥 → (∀𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))))
2517, 18, 24cbvexv1 2278 . . . 4 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)))
268, 13, 253imtr3i 283 . . 3 (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)))
277, 26bj-chvarvv 33580 . 2 (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
282719.35ri 1842 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
 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 5049 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-axrep2  33625
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