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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 )