Theorem bnj581

Description: Technical lemma for bnj580 . 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) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj581.3	⊢ ( 𝜒 ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj581.4	⊢ ( 𝜑′ ↔ [ 𝑔 / 𝑓 ] 𝜑 )
	bnj581.5	⊢ ( 𝜓′ ↔ [ 𝑔 / 𝑓 ] 𝜓 )
	bnj581.6	⊢ ( 𝜒′ ↔ [ 𝑔 / 𝑓 ] 𝜒 )
Assertion	bnj581	⊢ ( 𝜒′ ↔ ( 𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj581.3	⊢ ( 𝜒 ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
2		bnj581.4	⊢ ( 𝜑′ ↔ [ 𝑔 / 𝑓 ] 𝜑 )
3		bnj581.5	⊢ ( 𝜓′ ↔ [ 𝑔 / 𝑓 ] 𝜓 )
4		bnj581.6	⊢ ( 𝜒′ ↔ [ 𝑔 / 𝑓 ] 𝜒 )
5	1	sbcbii	⊢ ( [ 𝑔 / 𝑓 ] 𝜒 ↔ [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
6		sbc3an	⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 𝑛 ∧ [ 𝑔 / 𝑓 ] 𝜑 ∧ [ 𝑔 / 𝑓 ] 𝜓 ) )
7		bnj62	⊢ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛 )
8	7	bicomi	⊢ ( 𝑔 Fn 𝑛 ↔ [ 𝑔 / 𝑓 ] 𝑓 Fn 𝑛 )
9	8 2 3	3anbi123i	⊢ ( ( 𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 𝑛 ∧ [ 𝑔 / 𝑓 ] 𝜑 ∧ [ 𝑔 / 𝑓 ] 𝜓 ) )
10	6 9	bitr4i	⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )
11	4 5 10	3bitri	⊢ ( 𝜒′ ↔ ( 𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )