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Theorem ralxp 4422
 Description: Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1 (x = ⟨y, z⟩ → (φψ))
Assertion
Ref Expression
ralxp (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 ralxp
StepHypRef Expression
1 iunxpconst 4343 . . 3 y A ({y} × B) = (A × B)
21raleqi 2503 . 2 (x y A ({y} × B)φx (A × B)φ)
3 ralxp.1 . . 3 (x = ⟨y, z⟩ → (φψ))
43raliunxp 4420 . 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 1242  ∀wral 2300  {csn 3367  ⟨cop 3370  ∪ ciun 3648   × cxp 4286 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294  df-rel 4295 This theorem is referenced by:  ralxpf  4425  issref  4650  ffnov  5547  eqfnov  5549  funimassov  5592  f1stres  5728  f2ndres  5729  ecopover  6140  ecopoverg  6143
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