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Theorem cbv1h 1978
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbv1h.1 (φ → (ψyψ))
cbv1h.2 (φ → (χxχ))
cbv1h.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbv1h (xyφ → (xψyχ))

Proof of Theorem cbv1h
StepHypRef Expression
1 cbv1h.1 . . . . 5 (φ → (ψyψ))
21sps 1754 . . . 4 (yφ → (ψyψ))
32al2imi 1561 . . 3 (xyφ → (xψxyψ))
4 ax-7 1734 . . 3 (xyψyxψ)
53, 4syl6 29 . 2 (xyφ → (xψyxψ))
6 cbv1h.3 . . . . . . . 8 (φ → (x = y → (ψχ)))
76com23 72 . . . . . . 7 (φ → (ψ → (x = yχ)))
8 cbv1h.2 . . . . . . 7 (φ → (χxχ))
97, 8syl6d 64 . . . . . 6 (φ → (ψ → (x = yxχ)))
109al2imi 1561 . . . . 5 (xφ → (xψx(x = yxχ)))
11 ax9o 1950 . . . . 5 (x(x = yxχ) → χ)
1210, 11syl6 29 . . . 4 (xφ → (xψχ))
1312al2imi 1561 . . 3 (yxφ → (yxψyχ))
1413a7s 1735 . 2 (xyφ → (yxψyχ))
155, 14syld 40 1 (xyφ → (xψyχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 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:  cbv1  1979  cbv2h  1980  cbv3h  1983
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