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Theorem bm1.3ii 5205
Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5202. 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 1950 . . 3 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) ↔ (∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))))
2 bimsc1 840 . . . . 5 (((𝜑𝑦𝑧) ∧ (𝑦𝑥 ↔ (𝑦𝑧𝜑))) → (𝑦𝑥𝜑))
32alanimi 1813 . . . 4 ((∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∀𝑦(𝑦𝑥𝜑))
43eximi 1831 . . 3 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
51, 4sylbir 237 . 2 ((∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
6 bm1.3ii.1 . . . 4 𝑥𝑦(𝜑𝑦𝑥)
7 elequ2 2125 . . . . . . 7 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
87imbi2d 343 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑦𝑥) ↔ (𝜑𝑦𝑧)))
98albidv 1917 . . . . 5 (𝑥 = 𝑧 → (∀𝑦(𝜑𝑦𝑥) ↔ ∀𝑦(𝜑𝑦𝑧)))
109cbvexvw 2040 . . . 4 (∃𝑥𝑦(𝜑𝑦𝑥) ↔ ∃𝑧𝑦(𝜑𝑦𝑧))
116, 10mpbi 232 . . 3 𝑧𝑦(𝜑𝑦𝑧)
12 ax-sep 5202 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))
1311, 12exan 1858 . 2 𝑧(∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑)))
145, 13exlimiiv 1928 1 𝑥𝑦(𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1531  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-sep 5202
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777
This theorem is referenced by:  axpow3  5268  vpwex  5277  axprlem4  5326  zfpair2  5330  axun2  7462  uniex2  7463
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