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Theorem rexxpf 5711
Description: Version of rexxp 5706 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
ralxpf.1 𝑦𝜑
ralxpf.2 𝑧𝜑
ralxpf.3 𝑥𝜓
ralxpf.4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
rexxpf (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem rexxpf
StepHypRef Expression
1 ralxpf.1 . . . . . 6 𝑦𝜑
21nfn 1851 . . . . 5 𝑦 ¬ 𝜑
3 ralxpf.2 . . . . . 6 𝑧𝜑
43nfn 1851 . . . . 5 𝑧 ¬ 𝜑
5 ralxpf.3 . . . . . 6 𝑥𝜓
65nfn 1851 . . . . 5 𝑥 ¬ 𝜓
7 ralxpf.4 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
87notbid 320 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
92, 4, 6, 8ralxpf 5710 . . . 4 (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴𝑧𝐵 ¬ 𝜓)
10 ralnex 3234 . . . . 5 (∀𝑧𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧𝐵 𝜓)
1110ralbii 3163 . . . 4 (∀𝑦𝐴𝑧𝐵 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
129, 11bitri 277 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
1312notbii 322 . 2 (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
14 dfrex2 3237 . 2 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑)
15 dfrex2 3237 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
1613, 14, 153bitr4i 305 1 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208   = wceq 1531  wnf 1778  wral 3136  wrex 3137  cop 4565   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-iun 4912  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  iunxpf  5712  wdom2d2  39622
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