Metamath Proof Explorer


Theorem frpoins2g

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022)

Ref Expression
Hypotheses frpoins2g.1 ( 𝑦𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓𝜑 ) )
frpoins2g.3 ( 𝑦 = 𝑧 → ( 𝜑𝜓 ) )
Assertion frpoins2g ( ( 𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴 ) → ∀ 𝑦𝐴 𝜑 )

Proof

Step Hyp Ref Expression
1 frpoins2g.1 ( 𝑦𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓𝜑 ) )
2 frpoins2g.3 ( 𝑦 = 𝑧 → ( 𝜑𝜓 ) )
3 nfv 𝑦 𝜓
4 1 3 2 frpoins2fg ( ( 𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴 ) → ∀ 𝑦𝐴 𝜑 )