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Theorem ralxpf 4424
 Description: Version of ralxp 4421 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (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
ralxpf (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 ralxpf
Dummy variables v u w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2538 . 2 (x (A × B)φv (A × B)[v / x]φ)
2 cbvralsv 2538 . . . 4 (z B [w / y]ψu B [u / z][w / y]ψ)
32ralbii 2324 . . 3 (w A z B [w / y]ψw A u B [u / z][w / y]ψ)
4 nfv 1418 . . . 4 wz B ψ
5 nfcv 2175 . . . . 5 yB
6 nfs1v 1812 . . . . 5 y[w / y]ψ
75, 6nfralxy 2354 . . . 4 yz B [w / y]ψ
8 sbequ12 1651 . . . . 5 (y = w → (ψ ↔ [w / y]ψ))
98ralbidv 2320 . . . 4 (y = w → (z B ψz B [w / y]ψ))
104, 7, 9cbvral 2523 . . 3 (y A z B ψw A z B [w / y]ψ)
11 vex 2554 . . . . . 6 w V
12 vex 2554 . . . . . 6 u V
1311, 12eqvinop 3970 . . . . 5 (v = ⟨w, u⟩ ↔ yz(v = ⟨y, zy, z⟩ = ⟨w, u⟩))
14 ralxpf.1 . . . . . . . 8 yφ
1514nfsb 1819 . . . . . . 7 y[v / x]φ
166nfsb 1819 . . . . . . 7 y[u / z][w / y]ψ
1715, 16nfbi 1478 . . . . . 6 y([v / x]φ ↔ [u / z][w / y]ψ)
18 ralxpf.2 . . . . . . . . 9 zφ
1918nfsb 1819 . . . . . . . 8 z[v / x]φ
20 nfs1v 1812 . . . . . . . 8 z[u / z][w / y]ψ
2119, 20nfbi 1478 . . . . . . 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 2597 . . . . . . . 8 (v = ⟨y, z⟩ → ([v / x]φψ))
25 vex 2554 . . . . . . . . . 10 y V
26 vex 2554 . . . . . . . . . 10 z V
2725, 26opth 3964 . . . . . . . . 9 (⟨y, z⟩ = ⟨w, u⟩ ↔ (y = w z = u))
28 sbequ12 1651 . . . . . . . . . 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 1482 . . . . . 6 (z(v = ⟨y, zy, z⟩ = ⟨w, u⟩) → ([v / x]φ ↔ [u / z][w / y]ψ))
3317, 32exlimi 1482 . . . . 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))
3534ralxp 4421 . . 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 1242  Ⅎwnf 1346  ∃wex 1378  [wsb 1642  ∀wral 2300  ⟨cop 3369   × cxp 4285 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 3865  ax-pow 3917  ax-pr 3934 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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-iun 3649  df-opab 3809  df-xp 4293  df-rel 4294 This theorem is referenced by: (None)
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