Theorem frpoins2fg

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

	Ref	Expression
Hypotheses	frpoins2fg.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
	frpoins2fg.2	⊢ Ⅎ 𝑦 𝜓
	frpoins2fg.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	frpoins2fg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		frpoins2fg.1	⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
2		frpoins2fg.2	⊢ Ⅎ 𝑦 𝜓
3		frpoins2fg.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		sbsbc	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 )
5	2 3	sbiev	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 )
6	4 5	bitr3i	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 )
7	6	ralbii	⊢ ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 )
8	1	adantl	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) )
9	7 8	syl5bi	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) [ 𝑧 / 𝑦 ] 𝜑 → 𝜑 ) )
10	9	frpoinsg	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ 𝐴 𝜑 )