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Theorem cbvex2 2005
 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 1843 . 2 zyφ
3 cbval2.3 . . 3 xψ
43nfex 1843 . 2 xwψ
5 nfv 1619 . . . . . 6 w x = z
6 cbval2.2 . . . . . 6 wφ
75, 6nfan 1824 . . . . 5 w(x = z φ)
8 nfv 1619 . . . . . 6 y x = z
9 cbval2.4 . . . . . 6 yψ
108, 9nfan 1824 . . . . 5 y(x = z ψ)
11 cbval2.5 . . . . . . 7 ((x = z y = w) → (φψ))
1211expcom 424 . . . . . 6 (y = w → (x = z → (φψ)))
1312pm5.32d 620 . . . . 5 (y = w → ((x = z φ) ↔ (x = z ψ)))
147, 10, 13cbvex 1985 . . . 4 (y(x = z φ) ↔ w(x = z ψ))
15 19.42v 1905 . . . 4 (y(x = z φ) ↔ (x = z yφ))
16 19.42v 1905 . . . 4 (w(x = z ψ) ↔ (x = z wψ))
1714, 15, 163bitr3i 266 . . 3 ((x = z yφ) ↔ (x = z wψ))
18 pm5.32 617 . . 3 ((x = z → (yφwψ)) ↔ ((x = z yφ) ↔ (x = z wψ)))
1917, 18mpbir 200 . 2 (x = z → (yφwψ))
202, 4, 19cbvex 1985 1 (xyφzwψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  cbvex2v  2007  2eu6  2289  cbvopab  4630  cbvoprab12  5569
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