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Theorem raliunxp 4404
Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4406, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1 (x = ⟨y, z⟩ → (φψ))
Assertion
Ref Expression
raliunxp (x y A ({y} × B)φy A z B ψ)
Distinct variable groups:   x,y,z,A   x,B,z   φ,y,z   ψ,x
Allowed substitution hints:   φ(x)   ψ(y,z)   B(y)

Proof of Theorem raliunxp
StepHypRef Expression
1 eliunxp 4402 . . . . . 6 (x y A ({y} × B) ↔ yz(x = ⟨y, z (y A z B)))
21imbi1i 227 . . . . 5 ((x y A ({y} × B) → φ) ↔ (yz(x = ⟨y, z (y A z B)) → φ))
3 19.23vv 1746 . . . . 5 (yz((x = ⟨y, z (y A z B)) → φ) ↔ (yz(x = ⟨y, z (y A z B)) → φ))
42, 3bitr4i 176 . . . 4 ((x y A ({y} × B) → φ) ↔ yz((x = ⟨y, z (y A z B)) → φ))
54albii 1339 . . 3 (x(x y A ({y} × B) → φ) ↔ xyz((x = ⟨y, z (y A z B)) → φ))
6 alrot3 1354 . . . 4 (xyz((x = ⟨y, z (y A z B)) → φ) ↔ yzx((x = ⟨y, z (y A z B)) → φ))
7 impexp 250 . . . . . . 7 (((x = ⟨y, z (y A z B)) → φ) ↔ (x = ⟨y, z⟩ → ((y A z B) → φ)))
87albii 1339 . . . . . 6 (x((x = ⟨y, z (y A z B)) → φ) ↔ x(x = ⟨y, z⟩ → ((y A z B) → φ)))
9 vex 2538 . . . . . . . 8 y V
10 vex 2538 . . . . . . . 8 z V
119, 10opex 3940 . . . . . . 7 y, z V
12 ralxp.1 . . . . . . . 8 (x = ⟨y, z⟩ → (φψ))
1312imbi2d 219 . . . . . . 7 (x = ⟨y, z⟩ → (((y A z B) → φ) ↔ ((y A z B) → ψ)))
1411, 13ceqsalv 2561 . . . . . 6 (x(x = ⟨y, z⟩ → ((y A z B) → φ)) ↔ ((y A z B) → ψ))
158, 14bitri 173 . . . . 5 (x((x = ⟨y, z (y A z B)) → φ) ↔ ((y A z B) → ψ))
16152albii 1340 . . . 4 (yzx((x = ⟨y, z (y A z B)) → φ) ↔ yz((y A z B) → ψ))
176, 16bitri 173 . . 3 (xyz((x = ⟨y, z (y A z B)) → φ) ↔ yz((y A z B) → ψ))
185, 17bitri 173 . 2 (x(x y A ({y} × B) → φ) ↔ yz((y A z B) → ψ))
19 df-ral 2289 . 2 (x y A ({y} × B)φx(x y A ({y} × B) → φ))
20 r2al 2321 . 2 (y A z B ψyz((y A z B) → ψ))
2118, 19, 203bitr4i 201 1 (x y A ({y} × B)φy A z B ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362   wcel 1374  wral 2284  {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:  ralxp  4406  fmpt2x  5749
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