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Theorem bj-axrep4 32434
Description: Remove dependency on ax-13 2245 from axrep4 4735. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-axrep4.1 𝑧𝜑
Assertion
Ref Expression
bj-axrep4 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem bj-axrep4
StepHypRef Expression
1 bj-axrep3 32433 . . 3 𝑥(∃𝑧𝑦(𝜑𝑦 = 𝑧) → ∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
2119.35i 1803 . 2 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
3 nfv 1840 . . . . 5 𝑧 𝑦𝑥
4 nfv 1840 . . . . . . 7 𝑧 𝑥𝑤
5 nfa1 2025 . . . . . . 7 𝑧𝑧𝜑
64, 5nfan 1825 . . . . . 6 𝑧(𝑥𝑤 ∧ ∀𝑧𝜑)
76nfex 2151 . . . . 5 𝑧𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)
83, 7nfbi 1830 . . . 4 𝑧(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
98nfal 2150 . . 3 𝑧𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
10 nfv 1840 . . . . 5 𝑥 𝑦𝑧
11 nfe1 2024 . . . . 5 𝑥𝑥(𝑥𝑤𝜑)
1210, 11nfbi 1830 . . . 4 𝑥(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
1312nfal 2150 . . 3 𝑥𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
14 elequ2 2001 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
15 bj-axrep4.1 . . . . . . . . 9 𝑧𝜑
161519.3 2067 . . . . . . . 8 (∀𝑧𝜑𝜑)
1716anbi2i 729 . . . . . . 7 ((𝑥𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥𝑤𝜑))
1817exbii 1771 . . . . . 6 (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
1918a1i 11 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2014, 19bibi12d 335 . . . 4 (𝑥 = 𝑧 → ((𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
2120albidv 1846 . . 3 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
229, 13, 21cbvexv1 2175 . 2 (∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
232, 22sylib 208 1 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-rep 4731
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-axrep5  32435
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