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Theorem cbvex2 1794
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbval2.1 zφ
cbval2.2 wφ
cbval2.3 xψ
cbval2.4 yψ
cbval2.5 ((x = z y = w) → (φψ))
Assertion
Ref Expression
cbvex2 (xyφzwψ)
Distinct variable groups:   x,y   y,z   x,w   z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvex2
StepHypRef Expression
1 cbval2.1 . . 3 zφ
21nfex 1525 . 2 zyφ
3 cbval2.3 . . 3 xψ
43nfex 1525 . 2 xwψ
5 nfv 1418 . . . . . 6 w x = z
6 cbval2.2 . . . . . 6 wφ
75, 6nfan 1454 . . . . 5 w(x = z φ)
8 nfv 1418 . . . . . 6 y x = z
9 cbval2.4 . . . . . 6 yψ
108, 9nfan 1454 . . . . 5 y(x = z ψ)
11 cbval2.5 . . . . . . 7 ((x = z y = w) → (φψ))
1211expcom 109 . . . . . 6 (y = w → (x = z → (φψ)))
1312pm5.32d 423 . . . . 5 (y = w → ((x = z φ) ↔ (x = z ψ)))
147, 10, 13cbvex 1636 . . . 4 (y(x = z φ) ↔ w(x = z ψ))
15 19.42v 1783 . . . 4 (y(x = z φ) ↔ (x = z yφ))
16 19.42v 1783 . . . 4 (w(x = z ψ) ↔ (x = z wψ))
1714, 15, 163bitr3i 199 . . 3 ((x = z yφ) ↔ (x = z wψ))
18 pm5.32 426 . . 3 ((x = z → (yφwψ)) ↔ ((x = z yφ) ↔ (x = z wψ)))
1917, 18mpbir 134 . 2 (x = z → (yφwψ))
202, 4, 19cbvex 1636 1 (xyφzwψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wnf 1346  wex 1378
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  cbvex2v  1796  cbvopab  3819  cbvoprab12  5520
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