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Theorem r19.12 2439
 Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.12 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem r19.12
StepHypRef Expression
1 nfcv 2194 . . . 4 𝑦𝐴
2 nfra1 2372 . . . 4 𝑦𝑦𝐵 𝜑
31, 2nfrexxy 2378 . . 3 𝑦𝑥𝐴𝑦𝐵 𝜑
4 ax-1 5 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
53, 4ralrimi 2407 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑)
6 rsp 2386 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
76com12 30 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
87reximdv 2437 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
98ralimia 2399 . 2 (∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
105, 9syl 14 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329 This theorem is referenced by:  iuniin  3695
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