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Metamath Proof Explorer

Theorem bnj157

Description: Well-founded induction restricted to a set ( A e. _V ). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj157.1 ⊢ ψ ↔ ∀ y ∈ A y R x → [˙y / x]˙ φ
bnj157.2 ⊢ A ∈ V
bnj157.3 ⊢ R Fr A
Assertion bnj157 ⊢ ∀ x ∈ A ψ → φ → ∀ x ∈ A φ

Proof

Step Hyp Ref Expression
1 bnj157.1 ⊢ ψ ↔ ∀ y ∈ A y R x → [˙y / x]˙ φ
2 bnj157.2 ⊢ A ∈ V
3 bnj157.3 ⊢ R Fr A
4 2 1 bnj110 ⊢ R Fr A ∧ ∀ x ∈ A ψ → φ → ∀ x ∈ A φ
5 3 4 mpan ⊢ ∀ x ∈ A ψ → φ → ∀ x ∈ A φ