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Theorem bj-axsep 32433
 Description: Remove dependency on ax-13 2245 from axsep 4740. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axsep 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-axsep
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . 4 𝑦(𝑤 = 𝑥𝜑)
21bj-axrep5 32432 . . 3 (∀𝑤(𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)) → ∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))))
3 equtr 1945 . . . . . . . 8 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑦 = 𝑥))
4 equcomi 1941 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4syl6 35 . . . . . . 7 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑥 = 𝑦))
65adantrd 484 . . . . . 6 (𝑦 = 𝑤 → ((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
76alrimiv 1852 . . . . 5 (𝑦 = 𝑤 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
87a1d 25 . . . 4 (𝑦 = 𝑤 → (𝑤𝑧 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)))
98bj-spimevv 32361 . . 3 (𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
102, 9mpg 1721 . 2 𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
11 an12 837 . . . . . . 7 ((𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
1211exbii 1771 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
13 elequ1 1994 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
1413anbi1d 740 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤𝑧𝜑) ↔ (𝑥𝑧𝜑)))
1514equsexvw 1929 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑥𝑧𝜑))
1612, 15bitr3i 266 . . . . 5 (∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)) ↔ (𝑥𝑧𝜑))
1716bibi2i 327 . . . 4 ((𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1817albii 1744 . . 3 (∀𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1918exbii 1771 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
2010, 19mpbi 220 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701 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-8 1989  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: (None)
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