Database
CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
Axiom scheme ax-13 (Quantified Equality)
cbv1h
Metamath Proof Explorer
Description: Rule used to change bound variables, using implicit substitution. Usage
of this theorem is discouraged because it depends on ax-13 .
(Contributed by NM , 11-May-1993) (Proof shortened by Wolf Lammen , 13-May-2018) (New usage is discouraged.)
Ref
Expression
Hypotheses
cbv1h.1
⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
cbv1h.2
⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
cbv1h.3
⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
Assertion
cbv1h
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
Proof
Step
Hyp
Ref
Expression
1
cbv1h.1
⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
2
cbv1h.2
⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3
cbv1h.3
⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
4
nfa1
⊢ Ⅎ 𝑥 ∀ 𝑥 ∀ 𝑦 𝜑
5
nfa2
⊢ Ⅎ 𝑦 ∀ 𝑥 ∀ 𝑦 𝜑
6
2sp
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 )
7
6 1
syl
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
8
5 7
nf5d
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → Ⅎ 𝑦 𝜓 )
9
6 2
syl
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
10
4 9
nf5d
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → Ⅎ 𝑥 𝜒 )
11
6 3
syl
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
12
4 5 8 10 11
cbv1
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )