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Theorem cbv1h 1611
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
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 nfa1 1412 . 2 xxyφ
2 nfa2 1449 . 2 yxyφ
3 sp 1378 . . . . 5 (yφφ)
43sps 1408 . . . 4 (xyφφ)
5 cbv1h.1 . . . 4 (φ → (ψyψ))
64, 5syl 14 . . 3 (xyφ → (ψyψ))
72, 6nfd 1393 . 2 (xyφ → Ⅎyψ)
8 cbv1h.2 . . . 4 (φ → (χxχ))
94, 8syl 14 . . 3 (xyφ → (χxχ))
101, 9nfd 1393 . 2 (xyφ → Ⅎxχ)
11 cbv1h.3 . . 3 (φ → (x = y → (ψχ)))
124, 11syl 14 . 2 (xyφ → (x = y → (ψχ)))
131, 2, 7, 10, 12cbv1 1610 1 (xyφ → (xψyχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1224
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326
This theorem is referenced by:  cbv2h  1612
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