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Theorem cbvexdh 1783
Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1875. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
cbvexdh.1 (φyφ)
cbvexdh.2 (φ → (ψyψ))
cbvexdh.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbvexdh (φ → (xψyχ))
Distinct variable groups:   φ,x   χ,x
Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem cbvexdh
StepHypRef Expression
1 ax-17 1400 . . 3 (φxφ)
2 ax-17 1400 . . . 4 (χxχ)
32hbex 1509 . . 3 (yχxyχ)
4 cbvexdh.1 . . . . 5 (φyφ)
5 cbvexdh.2 . . . . 5 (φ → (ψyψ))
6 cbvexdh.3 . . . . . 6 (φ → (x = y → (ψχ)))
7 equcomi 1574 . . . . . . 7 (y = xx = y)
8 bicom1 122 . . . . . . 7 ((ψχ) → (χψ))
97, 8imim12i 53 . . . . . 6 ((x = y → (ψχ)) → (y = x → (χψ)))
106, 9syl 14 . . . . 5 (φ → (y = x → (χψ)))
114, 5, 10equsexd 1599 . . . 4 (φ → (y(y = x χ) ↔ ψ))
12 simpr 103 . . . . 5 ((y = x χ) → χ)
1312eximi 1473 . . . 4 (y(y = x χ) → yχ)
1411, 13syl6bir 153 . . 3 (φ → (ψyχ))
151, 3, 14exlimdh 1469 . 2 (φ → (xψyχ))
161, 5eximdh 1484 . . . 4 (φ → (xψxyψ))
17 19.12 1537 . . . 4 (xyψyxψ)
1816, 17syl6 29 . . 3 (φ → (xψyxψ))
192a1i 9 . . . . 5 (φ → (χxχ))
201, 19, 6equsexd 1599 . . . 4 (φ → (x(x = y ψ) ↔ χ))
21 simpr 103 . . . . 5 ((x = y ψ) → ψ)
2221eximi 1473 . . . 4 (x(x = y ψ) → xψ)
2320, 22syl6bir 153 . . 3 (φ → (χxψ))
244, 18, 23exlimd2 1468 . 2 (φ → (yχxψ))
2515, 24impbid 120 1 (φ → (xψyχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  cbvexd  1784
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