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Theorem cbvralf 2501
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1 xA
cbvralf.2 yA
cbvralf.3 yφ
cbvralf.4 xψ
cbvralf.5 (x = y → (φψ))
Assertion
Ref Expression
cbvralf (x A φy A ψ)

Proof of Theorem cbvralf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1398 . . . 4 z(x Aφ)
2 cbvralf.1 . . . . . 6 xA
32nfcri 2150 . . . . 5 x z A
4 nfs1v 1793 . . . . 5 x[z / x]φ
53, 4nfim 1442 . . . 4 x(z A → [z / x]φ)
6 eleq1 2078 . . . . 5 (x = z → (x Az A))
7 sbequ12 1632 . . . . 5 (x = z → (φ ↔ [z / x]φ))
86, 7imbi12d 223 . . . 4 (x = z → ((x Aφ) ↔ (z A → [z / x]φ)))
91, 5, 8cbval 1615 . . 3 (x(x Aφ) ↔ z(z A → [z / x]φ))
10 cbvralf.2 . . . . . 6 yA
1110nfcri 2150 . . . . 5 y z A
12 cbvralf.3 . . . . . 6 yφ
1312nfsb 1800 . . . . 5 y[z / x]φ
1411, 13nfim 1442 . . . 4 y(z A → [z / x]φ)
15 nfv 1398 . . . 4 z(y Aψ)
16 eleq1 2078 . . . . 5 (z = y → (z Ay A))
17 sbequ 1699 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
18 cbvralf.4 . . . . . . 7 xψ
19 cbvralf.5 . . . . . . 7 (x = y → (φψ))
2018, 19sbie 1652 . . . . . 6 ([y / x]φψ)
2117, 20syl6bb 185 . . . . 5 (z = y → ([z / x]φψ))
2216, 21imbi12d 223 . . . 4 (z = y → ((z A → [z / x]φ) ↔ (y Aψ)))
2314, 15, 22cbval 1615 . . 3 (z(z A → [z / x]φ) ↔ y(y Aψ))
249, 23bitri 173 . 2 (x(x Aφ) ↔ y(y Aψ))
25 df-ral 2285 . 2 (x A φx(x Aφ))
26 df-ral 2285 . 2 (y A ψy(y Aψ))
2724, 25, 263bitr4i 201 1 (x A φy A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1224  Ⅎwnf 1325   ∈ wcel 1370  [wsb 1623  Ⅎwnfc 2143  ∀wral 2280 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285 This theorem is referenced by:  cbvral  2503  ffnfvf  5245
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