Theorem bnj985

Description: Technical lemma for bnj69 . 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). See bnj985v for a version with more disjoint variable conditions, not requiring ax-13 . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj985.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj985.6	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
	bnj985.9	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
	bnj985.11	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj985.13	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
Assertion	bnj985	⊢ ( 𝐺 ∈ 𝐵 ↔ ∃ 𝑝 𝜒″ )

Proof

Step	Hyp	Ref	Expression
1		bnj985.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
2		bnj985.6	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
3		bnj985.9	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
4		bnj985.11	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
5		bnj985.13	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
6	5	bnj918	⊢ 𝐺 ∈ V
7	1 4	bnj984	⊢ ( 𝐺 ∈ V → ( 𝐺 ∈ 𝐵 ↔ [ 𝐺 / 𝑓 ] ∃ 𝑛 𝜒 ) )
8	6 7	ax-mp	⊢ ( 𝐺 ∈ 𝐵 ↔ [ 𝐺 / 𝑓 ] ∃ 𝑛 𝜒 )
9		sbcex2	⊢ ( [ 𝐺 / 𝑓 ] ∃ 𝑝 𝜒′ ↔ ∃ 𝑝 [ 𝐺 / 𝑓 ] 𝜒′ )
10		nfv	⊢ Ⅎ 𝑝 𝜒
11	10	sb8e	⊢ ( ∃ 𝑛 𝜒 ↔ ∃ 𝑝 [ 𝑝 / 𝑛 ] 𝜒 )
12		sbsbc	⊢ ( [ 𝑝 / 𝑛 ] 𝜒 ↔ [ 𝑝 / 𝑛 ] 𝜒 )
13	12	exbii	⊢ ( ∃ 𝑝 [ 𝑝 / 𝑛 ] 𝜒 ↔ ∃ 𝑝 [ 𝑝 / 𝑛 ] 𝜒 )
14	11 13	bitri	⊢ ( ∃ 𝑛 𝜒 ↔ ∃ 𝑝 [ 𝑝 / 𝑛 ] 𝜒 )
15	14 2	bnj133	⊢ ( ∃ 𝑛 𝜒 ↔ ∃ 𝑝 𝜒′ )
16	15	sbcbii	⊢ ( [ 𝐺 / 𝑓 ] ∃ 𝑛 𝜒 ↔ [ 𝐺 / 𝑓 ] ∃ 𝑝 𝜒′ )
17	3	exbii	⊢ ( ∃ 𝑝 𝜒″ ↔ ∃ 𝑝 [ 𝐺 / 𝑓 ] 𝜒′ )
18	9 16 17	3bitr4i	⊢ ( [ 𝐺 / 𝑓 ] ∃ 𝑛 𝜒 ↔ ∃ 𝑝 𝜒″ )
19	8 18	bitri	⊢ ( 𝐺 ∈ 𝐵 ↔ ∃ 𝑝 𝜒″ )