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Theorem cbv1h 1677
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 → (𝜓 → ∀𝑦𝜓))
cbv1h.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv1h.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1h (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1h
StepHypRef Expression
1 nfa1 1477 . 2 𝑥𝑥𝑦𝜑
2 nfa2 1514 . 2 𝑦𝑥𝑦𝜑
3 sp 1444 . . . . 5 (∀𝑦𝜑𝜑)
43sps 1473 . . . 4 (∀𝑥𝑦𝜑𝜑)
5 cbv1h.1 . . . 4 (𝜑 → (𝜓 → ∀𝑦𝜓))
64, 5syl 14 . . 3 (∀𝑥𝑦𝜑 → (𝜓 → ∀𝑦𝜓))
72, 6nfd 1459 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜓)
8 cbv1h.2 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
94, 8syl 14 . . 3 (∀𝑥𝑦𝜑 → (𝜒 → ∀𝑥𝜒))
101, 9nfd 1459 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥𝜒)
11 cbv1h.3 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
124, 11syl 14 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
131, 2, 7, 10, 12cbv1 1676 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  cbv2h  1678
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