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Theorem rexxpf 5179
Description: Version of rexxp 5174 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 1767 . . . . 5 𝑦 ¬ 𝜑
3 ralxpf.2 . . . . . 6 𝑧𝜑
43nfn 1767 . . . . 5 𝑧 ¬ 𝜑
5 ralxpf.3 . . . . . 6 𝑥𝜓
65nfn 1767 . . . . 5 𝑥 ¬ 𝜓
7 ralxpf.4 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
87notbid 306 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
92, 4, 6, 8ralxpf 5178 . . . 4 (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴𝑧𝐵 ¬ 𝜓)
10 ralnex 2974 . . . . 5 (∀𝑧𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧𝐵 𝜓)
1110ralbii 2962 . . . 4 (∀𝑦𝐴𝑧𝐵 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
129, 11bitri 262 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
1312notbii 308 . 2 (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
14 dfrex2 2978 . 2 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑)
15 dfrex2 2978 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
1613, 14, 153bitr4i 290 1 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wnf 1698  wral 2895  wrex 2896  cop 4130   × cxp 5026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-iun 4451  df-opab 4638  df-xp 5034  df-rel 5035
This theorem is referenced by:  iunxpf  5180  wdom2d2  36416
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