Theorem nfopd

Description: Deduction version of bound-variable hypothesis builder nfop . This shows how the deduction version of a not-free theorem such as nfop can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008)

	Ref	Expression
Hypotheses	nfopd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfopd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion	nfopd	⊢ ( 𝜑 → Ⅎ 𝑥 ⟨ 𝐴 , 𝐵 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		nfopd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
2		nfopd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
3		nfaba1	⊢ Ⅎ 𝑥 { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 }
4		nfaba1	⊢ Ⅎ 𝑥 { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 }
5	3 4	nfop	⊢ Ⅎ 𝑥 ⟨ { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } , { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } ⟩
6		nfnfc1	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝐴
7		nfnfc1	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝐵
8	6 7	nfan	⊢ Ⅎ 𝑥 ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝐵 )
9		abidnf	⊢ ( Ⅎ 𝑥 𝐴 → { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } = 𝐴 )
10	9	adantr	⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝐵 ) → { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } = 𝐴 )
11		abidnf	⊢ ( Ⅎ 𝑥 𝐵 → { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } = 𝐵 )
12	11	adantl	⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝐵 ) → { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } = 𝐵 )
13	10 12	opeq12d	⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝐵 ) → ⟨ { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } , { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
14	8 13	nfceqdf	⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝐵 ) → ( Ⅎ 𝑥 ⟨ { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } , { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } ⟩ ↔ Ⅎ 𝑥 ⟨ 𝐴 , 𝐵 ⟩ ) )
15	1 2 14	syl2anc	⊢ ( 𝜑 → ( Ⅎ 𝑥 ⟨ { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐴 } , { 𝑧 ∣ ∀ 𝑥 𝑧 ∈ 𝐵 } ⟩ ↔ Ⅎ 𝑥 ⟨ 𝐴 , 𝐵 ⟩ ) )
16	5 15	mpbii	⊢ ( 𝜑 → Ⅎ 𝑥 ⟨ 𝐴 , 𝐵 ⟩ )