Theorem bnj124

Description: Technical lemma for bnj150 . 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) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj124.1	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
	bnj124.2	⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
	bnj124.3	⊢ ( 𝜓″ ↔ [ 𝐹 / 𝑓 ] 𝜓′ )
	bnj124.4	⊢ ( 𝜁″ ↔ [ 𝐹 / 𝑓 ] 𝜁′ )
	bnj124.5	⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
Assertion	bnj124	⊢ ( 𝜁″ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj124.1	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
2		bnj124.2	⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
3		bnj124.3	⊢ ( 𝜓″ ↔ [ 𝐹 / 𝑓 ] 𝜓′ )
4		bnj124.4	⊢ ( 𝜁″ ↔ [ 𝐹 / 𝑓 ] 𝜁′ )
5		bnj124.5	⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
6	5	sbcbii	⊢ ( [ 𝐹 / 𝑓 ] 𝜁′ ↔ [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
7	1	bnj95	⊢ 𝐹 ∈ V
8		nfv	⊢ Ⅎ 𝑓 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 )
9	8	sbc19.21g	⊢ ( 𝐹 ∈ V → ( [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ) )
10	7 9	ax-mp	⊢ ( [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
11		fneq1	⊢ ( 𝑓 = 𝑧 → ( 𝑓 Fn 1o ↔ 𝑧 Fn 1o ) )
12		fneq1	⊢ ( 𝑧 = 𝐹 → ( 𝑧 Fn 1o ↔ 𝐹 Fn 1o ) )
13	11 12	sbcie2g	⊢ ( 𝐹 ∈ V → ( [ 𝐹 / 𝑓 ] 𝑓 Fn 1o ↔ 𝐹 Fn 1o ) )
14	7 13	ax-mp	⊢ ( [ 𝐹 / 𝑓 ] 𝑓 Fn 1o ↔ 𝐹 Fn 1o )
15	14	bicomi	⊢ ( 𝐹 Fn 1o ↔ [ 𝐹 / 𝑓 ] 𝑓 Fn 1o )
16	15 2 3 7	bnj206	⊢ ( [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) )
17	16	imbi2i	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )
18	6 10 17	3bitri	⊢ ( [ 𝐹 / 𝑓 ] 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )
19	4 18	bitri	⊢ ( 𝜁″ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )