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Theorem cbval2 1793
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
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
cbval2 (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 cbval2
StepHypRef Expression
1 cbval2.1 . . 3 zφ
21nfal 1465 . 2 zyφ
3 cbval2.3 . . 3 xψ
43nfal 1465 . 2 xwψ
5 nfv 1418 . . . . . 6 w x = z
6 cbval2.2 . . . . . 6 wφ
75, 6nfim 1461 . . . . 5 w(x = zφ)
8 nfv 1418 . . . . . 6 y x = z
9 cbval2.4 . . . . . 6 yψ
108, 9nfim 1461 . . . . 5 y(x = zψ)
11 cbval2.5 . . . . . . 7 ((x = z y = w) → (φψ))
1211expcom 109 . . . . . 6 (y = w → (x = z → (φψ)))
1312pm5.74d 171 . . . . 5 (y = w → ((x = zφ) ↔ (x = zψ)))
147, 10, 13cbval 1634 . . . 4 (y(x = zφ) ↔ w(x = zψ))
15 19.21v 1750 . . . 4 (y(x = zφ) ↔ (x = zyφ))
16 19.21v 1750 . . . 4 (w(x = zψ) ↔ (x = zwψ))
1714, 15, 163bitr3i 199 . . 3 ((x = zyφ) ↔ (x = zwψ))
1817pm5.74ri 170 . 2 (x = z → (yφwψ))
192, 4, 18cbval 1634 1 (xyφzwψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346 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  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347 This theorem is referenced by:  cbval2v  1795
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