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Theorem r19.12 2576
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 2312 . . . 4 𝑦𝐴
2 nfra1 2501 . . . 4 𝑦𝑦𝐵 𝜑
31, 2nfrexxy 2509 . . 3 𝑦𝑥𝐴𝑦𝐵 𝜑
4 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
53, 4ralrimi 2541 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑)
6 rsp 2517 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
76com12 30 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
87reximdv 2571 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
98ralimia 2531 . 2 (∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
105, 9syl 14 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  wral 2448  wrex 2449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454
This theorem is referenced by:  iuniin  3881
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