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Theorem cbvrald 9196
Description: Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
Hypotheses
Ref Expression
cbvrald.nf0 xφ
cbvrald.nf1 yφ
cbvrald.nf2 (φ → Ⅎyψ)
cbvrald.nf3 (φ → Ⅎxχ)
cbvrald.is (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbvrald (φ → (x A ψy A χ))
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)

Proof of Theorem cbvrald
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbvrald.nf0 . . . 4 xφ
2 nfv 1418 . . . 4 zφ
3 nfv 1418 . . . . . 6 z x A
43a1i 9 . . . . 5 (φ → Ⅎz x A)
5 nfv 1418 . . . . . 6 zψ
65a1i 9 . . . . 5 (φ → Ⅎzψ)
74, 6nfimd 1474 . . . 4 (φ → Ⅎz(x Aψ))
8 nfv 1418 . . . . . 6 x z A
98a1i 9 . . . . 5 (φ → Ⅎx z A)
10 nfs1v 1812 . . . . . 6 x[z / x]ψ
1110a1i 9 . . . . 5 (φ → Ⅎx[z / x]ψ)
129, 11nfimd 1474 . . . 4 (φ → Ⅎx(z A → [z / x]ψ))
13 eleq1 2097 . . . . . . 7 (x = z → (x Az A))
1413adantl 262 . . . . . 6 ((φ x = z) → (x Az A))
15 sbequ12 1651 . . . . . . 7 (x = z → (ψ ↔ [z / x]ψ))
1615adantl 262 . . . . . 6 ((φ x = z) → (ψ ↔ [z / x]ψ))
1714, 16imbi12d 223 . . . . 5 ((φ x = z) → ((x Aψ) ↔ (z A → [z / x]ψ)))
1817ex 108 . . . 4 (φ → (x = z → ((x Aψ) ↔ (z A → [z / x]ψ))))
191, 2, 7, 12, 18cbv2 1632 . . 3 (φ → (x(x Aψ) ↔ z(z A → [z / x]ψ)))
20 cbvrald.nf1 . . . 4 yφ
21 nfv 1418 . . . . . 6 y z A
2221a1i 9 . . . . 5 (φ → Ⅎy z A)
23 cbvrald.nf2 . . . . . 6 (φ → Ⅎyψ)
241, 23nfsbd 1848 . . . . 5 (φ → Ⅎy[z / x]ψ)
2522, 24nfimd 1474 . . . 4 (φ → Ⅎy(z A → [z / x]ψ))
26 nfv 1418 . . . . . 6 z y A
2726a1i 9 . . . . 5 (φ → Ⅎz y A)
28 nfv 1418 . . . . . 6 zχ
2928a1i 9 . . . . 5 (φ → Ⅎzχ)
3027, 29nfimd 1474 . . . 4 (φ → Ⅎz(y Aχ))
31 eleq1 2097 . . . . . . 7 (z = y → (z Ay A))
3231adantl 262 . . . . . 6 ((φ z = y) → (z Ay A))
33 sbequ 1718 . . . . . . 7 (z = y → ([z / x]ψ ↔ [y / x]ψ))
34 cbvrald.nf3 . . . . . . . 8 (φ → Ⅎxχ)
35 cbvrald.is . . . . . . . 8 (φ → (x = y → (ψχ)))
361, 34, 35sbied 1668 . . . . . . 7 (φ → ([y / x]ψχ))
3733, 36sylan9bbr 436 . . . . . 6 ((φ z = y) → ([z / x]ψχ))
3832, 37imbi12d 223 . . . . 5 ((φ z = y) → ((z A → [z / x]ψ) ↔ (y Aχ)))
3938ex 108 . . . 4 (φ → (z = y → ((z A → [z / x]ψ) ↔ (y Aχ))))
402, 20, 25, 30, 39cbv2 1632 . . 3 (φ → (z(z A → [z / x]ψ) ↔ y(y Aχ)))
4119, 40bitrd 177 . 2 (φ → (x(x Aψ) ↔ y(y Aχ)))
42 df-ral 2305 . 2 (x A ψx(x Aψ))
43 df-ral 2305 . 2 (y A χy(y Aχ))
4441, 42, 433bitr4g 212 1 (φ → (x A ψy A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346   wcel 1390  [wsb 1642  wral 2300
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-ral 2305
This theorem is referenced by:  setindft  9349
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