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Theorem r2exf 2475
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1 𝑦𝐴
Assertion
Ref Expression
r2exf (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2exf
StepHypRef Expression
1 df-rex 2441 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 r2alf.1 . . . . . 6 𝑦𝐴
32nfcri 2293 . . . . 5 𝑦 𝑥𝐴
4319.42 1668 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
5 anass 399 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
65exbii 1585 . . . 4 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 df-rex 2441 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
87anbi2i 453 . . . 4 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
94, 6, 83bitr4i 211 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
109exbii 1585 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
111, 10bitr4i 186 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1472  wcel 2128  wnfc 2286  wrex 2436
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  r2ex  2477  rexcomf  2619
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