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Theorem cbv3 1982
 Description: Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2142. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbv3.1 yφ
cbv3.2 xψ
cbv3.3 (x = y → (φψ))
Assertion
Ref Expression
cbv3 (xφyψ)

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . . 4 yφ
21a1i 10 . . 3 ( ⊤ → Ⅎyφ)
3 cbv3.2 . . . 4 xψ
43a1i 10 . . 3 ( ⊤ → Ⅎxψ)
5 cbv3.3 . . . 4 (x = y → (φψ))
65a1i 10 . . 3 ( ⊤ → (x = y → (φψ)))
72, 4, 6cbv1 1979 . 2 (xy ⊤ → (xφyψ))
8 tru 1321 . . 3
98ax-gen 1546 . 2 y
107, 9mpg 1548 1 (xφyψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊤ wtru 1316  ∀wal 1540  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:  cbval  1984  ax16i  2046  ax16ALT2  2048  sb8  2092  mo  2226
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