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Theorem bm1.3ii 3919
Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3916. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-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 bm1.3ii.1 . . . . 5 𝑥𝑦(𝜑𝑦𝑥)
2 elequ2 1643 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
32imbi2d 228 . . . . . . 7 (𝑥 = 𝑧 → ((𝜑𝑦𝑥) ↔ (𝜑𝑦𝑧)))
43albidv 1747 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝜑𝑦𝑥) ↔ ∀𝑦(𝜑𝑦𝑧)))
54cbvexv 1838 . . . . 5 (∃𝑥𝑦(𝜑𝑦𝑥) ↔ ∃𝑧𝑦(𝜑𝑦𝑧))
61, 5mpbi 143 . . . 4 𝑧𝑦(𝜑𝑦𝑧)
7 ax-sep 3916 . . . 4 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))
86, 7pm3.2i 266 . . 3 (∃𝑧𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑)))
98exan 1624 . 2 𝑧(∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑)))
10 19.42v 1829 . . . 4 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) ↔ (∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))))
11 bimsc1 905 . . . . . 6 (((𝜑𝑦𝑧) ∧ (𝑦𝑥 ↔ (𝑦𝑧𝜑))) → (𝑦𝑥𝜑))
1211alanimi 1389 . . . . 5 ((∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∀𝑦(𝑦𝑥𝜑))
1312eximi 1532 . . . 4 (∃𝑥(∀𝑦(𝜑𝑦𝑧) ∧ ∀𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
1410, 13sylbir 133 . . 3 ((∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
1514exlimiv 1530 . 2 (∃𝑧(∀𝑦(𝜑𝑦𝑧) ∧ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧𝜑))) → ∃𝑥𝑦(𝑦𝑥𝜑))
169, 15ax-mp 7 1 𝑥𝑦(𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-sep 3916
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  axpow3  3971  pwex  3973  zfpair2  3993  axun2  4218  uniex2  4219
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