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Theorem rexxp 4407
Description: Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
ralxp.1 (x = ⟨y, z⟩ → (φψ))
Assertion
Ref Expression
rexxp (x (A × B)φy A z B ψ)
Distinct variable groups:   x,y,z,A   x,B,z   φ,y,z   ψ,x   y,B
Allowed substitution hints:   φ(x)   ψ(y,z)

Proof of Theorem rexxp
StepHypRef Expression
1 iunxpconst 4327 . . 3 y A ({y} × B) = (A × B)
21rexeqi 2488 . 2 (x y A ({y} × B)φx (A × B)φ)
3 ralxp.1 . . 3 (x = ⟨y, z⟩ → (φψ))
43rexiunxp 4405 . 2 (x y A ({y} × B)φy A z B ψ)
52, 4bitr3i 175 1 (x (A × B)φy A z B ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  wrex 2285  {csn 3350  cop 3353   ciun 3631   × cxp 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-xp 4278  df-rel 4279
This theorem is referenced by:  rexxpf  4410  fnrnov  5569  foov  5570  ovelimab  5574
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