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Theorem rexxpf 4399
Description: Version of rexxp 4396 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)φy A z B 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
Dummy variables v u w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvrexsv 2515 . 2 (x (A × B)φv (A × B)[v / x]φ)
2 cbvrexsv 2515 . . . 4 (z B [w / y]ψu B [u / z][w / y]ψ)
32rexbii 2301 . . 3 (w A z B [w / y]ψw A u B [u / z][w / y]ψ)
4 nfv 1395 . . . 4 wz B ψ
5 nfcv 2152 . . . . 5 yB
6 nfs1v 1789 . . . . 5 y[w / y]ψ
75, 6nfrexxy 2331 . . . 4 yz B [w / y]ψ
8 sbequ12 1628 . . . . 5 (y = w → (ψ ↔ [w / y]ψ))
98rexbidv 2297 . . . 4 (y = w → (z B ψz B [w / y]ψ))
104, 7, 9cbvrex 2500 . . 3 (y A z B ψw A z B [w / y]ψ)
11 vex 2530 . . . . . 6 w V
12 vex 2530 . . . . . 6 u V
1311, 12eqvinop 3944 . . . . 5 (v = ⟨w, u⟩ ↔ yz(v = ⟨y, zy, z⟩ = ⟨w, u⟩))
14 ralxpf.1 . . . . . . . 8 yφ
1514nfsb 1796 . . . . . . 7 y[v / x]φ
166nfsb 1796 . . . . . . 7 y[u / z][w / y]ψ
1715, 16nfbi 1455 . . . . . 6 y([v / x]φ ↔ [u / z][w / y]ψ)
18 ralxpf.2 . . . . . . . . 9 zφ
1918nfsb 1796 . . . . . . . 8 z[v / x]φ
20 nfs1v 1789 . . . . . . . 8 z[u / z][w / y]ψ
2119, 20nfbi 1455 . . . . . . 7 z([v / x]φ ↔ [u / z][w / y]ψ)
22 ralxpf.3 . . . . . . . . 9 xψ
23 ralxpf.4 . . . . . . . . 9 (x = ⟨y, z⟩ → (φψ))
2422, 23sbhypf 2572 . . . . . . . 8 (v = ⟨y, z⟩ → ([v / x]φψ))
25 vex 2530 . . . . . . . . . 10 y V
26 vex 2530 . . . . . . . . . 10 z V
2725, 26opth 3938 . . . . . . . . 9 (⟨y, z⟩ = ⟨w, u⟩ ↔ (y = w z = u))
28 sbequ12 1628 . . . . . . . . . 10 (z = u → ([w / y]ψ ↔ [u / z][w / y]ψ))
298, 28sylan9bb 435 . . . . . . . . 9 ((y = w z = u) → (ψ ↔ [u / z][w / y]ψ))
3027, 29sylbi 114 . . . . . . . 8 (⟨y, z⟩ = ⟨w, u⟩ → (ψ ↔ [u / z][w / y]ψ))
3124, 30sylan9bb 435 . . . . . . 7 ((v = ⟨y, zy, z⟩ = ⟨w, u⟩) → ([v / x]φ ↔ [u / z][w / y]ψ))
3221, 31exlimi 1459 . . . . . 6 (z(v = ⟨y, zy, z⟩ = ⟨w, u⟩) → ([v / x]φ ↔ [u / z][w / y]ψ))
3317, 32exlimi 1459 . . . . 5 (yz(v = ⟨y, zy, z⟩ = ⟨w, u⟩) → ([v / x]φ ↔ [u / z][w / y]ψ))
3413, 33sylbi 114 . . . 4 (v = ⟨w, u⟩ → ([v / x]φ ↔ [u / z][w / y]ψ))
3534rexxp 4396 . . 3 (v (A × B)[v / x]φw A u B [u / z][w / y]ψ)
363, 10, 353bitr4ri 202 . 2 (v (A × B)[v / x]φy A z B ψ)
371, 36bitri 173 1 (x (A × B)φy A z B ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1224  wnf 1323  wex 1355  [wsb 1619  wrex 2277  cop 3343   × cxp 4259
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-sbc 2734  df-csb 2822  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-iun 3623  df-opab 3783  df-xp 4267  df-rel 4268
This theorem is referenced by:  iunxpf  4400
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