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Theorem bm1.3ii 5010
 Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5007. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
bm1.3ii.1 𝑥𝑦(𝜑𝑦𝑥)
Assertion
Ref Expression
bm1.3ii 𝑥𝑦(𝑦𝑥𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bm1.3ii
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 19.42v 2052 . . 3 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) ↔ (∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))))
2 bimsc1 875 . . . . 5 (((𝜑𝑦𝑧) ∧ (𝑦𝑥 ↔ (𝑦𝑧𝜑))) → (𝑦𝑥𝜑))
32alanimi 1915 . . . 4 ((∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∀𝑦(𝑦𝑥𝜑))
43eximi 1933 . . 3 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
51, 4sylbir 227 . 2 ((∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
6 bm1.3ii.1 . . . . 5 𝑥𝑦(𝜑𝑦𝑥)
7 elequ2 2178 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
87imbi2d 332 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑦𝑥) ↔ (𝜑𝑦𝑧)))
98albidv 2019 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝜑𝑦𝑥) ↔ ∀𝑦(𝜑𝑦𝑧)))
109cbvexvw 2144 . . . . 5 (∃𝑥𝑦(𝜑𝑦𝑥) ↔ ∃𝑧𝑦(𝜑𝑦𝑧))
116, 10mpbi 222 . . . 4 𝑧𝑦(𝜑𝑦𝑧)
12 ax-sep 5007 . . . 4 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))
1311, 12pm3.2i 464 . . 3 (∃𝑧𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑)))
1413exan 1961 . 2 𝑧(∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑)))
155, 14exlimiiv 2030 1 𝑥𝑦(𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-sep 5007 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879 This theorem is referenced by:  axpow3  5070  vpwex  5079  zfpair2  5130  axun2  7216  uniex2  7217
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