Theorem bnj562

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj562.18	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
	bnj562.19	⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
	bnj562.38	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝜑″ )
Assertion	bnj562	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ) → 𝜑″ )

Proof

Step	Hyp	Ref	Expression
1		bnj562.18	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
2		bnj562.19	⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
3		bnj562.38	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝜑″ )
4	1 2	bnj556	⊢ ( 𝜂 → 𝜎 )
5	4 3	syl3an3	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ) → 𝜑″ )