New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbvralf GIF version

Theorem cbvralf 2829
 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 1619 . . . 4 z(x Aφ)
2 cbvralf.1 . . . . . 6 xA
32nfcri 2483 . . . . 5 x z A
4 nfs1v 2106 . . . . 5 x[z / x]φ
53, 4nfim 1813 . . . 4 x(z A → [z / x]φ)
6 eleq1 2413 . . . . 5 (x = z → (x Az A))
7 sbequ12 1919 . . . . 5 (x = z → (φ ↔ [z / x]φ))
86, 7imbi12d 311 . . . 4 (x = z → ((x Aφ) ↔ (z A → [z / x]φ)))
91, 5, 8cbval 1984 . . 3 (x(x Aφ) ↔ z(z A → [z / x]φ))
10 cbvralf.2 . . . . . 6 yA
1110nfcri 2483 . . . . 5 y z A
12 cbvralf.3 . . . . . 6 yφ
1312nfsb 2109 . . . . 5 y[z / x]φ
1411, 13nfim 1813 . . . 4 y(z A → [z / x]φ)
15 nfv 1619 . . . 4 z(y Aψ)
16 eleq1 2413 . . . . 5 (z = y → (z Ay A))
17 sbequ 2060 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
18 cbvralf.4 . . . . . . 7 xψ
19 cbvralf.5 . . . . . . 7 (x = y → (φψ))
2018, 19sbie 2038 . . . . . 6 ([y / x]φψ)
2117, 20syl6bb 252 . . . . 5 (z = y → ([z / x]φψ))
2216, 21imbi12d 311 . . . 4 (z = y → ((z A → [z / x]φ) ↔ (y Aψ)))
2314, 15, 22cbval 1984 . . 3 (z(z A → [z / x]φ) ↔ y(y Aψ))
249, 23bitri 240 . 2 (x(x Aφ) ↔ y(y Aψ))
25 df-ral 2619 . 2 (x A φx(x Aφ))
26 df-ral 2619 . 2 (y A ψy(y Aψ))
2724, 25, 263bitr4i 268 1 (x A φy A ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by:  cbvrexf  2830  cbvral  2831  ffnfvf  5428
 Copyright terms: Public domain W3C validator