copsex2d

Description: Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023)

	Ref	Expression
Hypotheses	copsex2d.xph	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	copsex2d.yph	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	copsex2d.xch	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
	copsex2d.ych	⊢ ( 𝜑 → Ⅎ 𝑦 𝜒 )
	copsex2d.exa	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	copsex2d.exb	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
	copsex2d.is	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	copsex2d	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2d.xph	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		copsex2d.yph	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		copsex2d.xch	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
4		copsex2d.ych	⊢ ( 𝜑 → Ⅎ 𝑦 𝜒 )
5		copsex2d.exa	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
6		copsex2d.exb	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
7		copsex2d.is	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
8		elisset	⊢ ( 𝐴 ∈ 𝑈 → ∃ 𝑥 𝑥 = 𝐴 )
9	5 8	syl	⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝐴 )
10		elisset	⊢ ( 𝐵 ∈ 𝑉 → ∃ 𝑦 𝑦 = 𝐵 )
11	6 10	syl	⊢ ( 𝜑 → ∃ 𝑦 𝑦 = 𝐵 )
12		exdistrv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
13		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 )
14	13	a1i	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
15	14 3	nfbid	⊢ ( 𝜑 → Ⅎ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
16	15	19.9d	⊢ ( 𝜑 → ( ∃ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
17		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 )
18	17	a1i	⊢ ( 𝜑 → Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
19	1 18	bj-nfexd	⊢ ( 𝜑 → Ⅎ 𝑦 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
20	19 4	nfbid	⊢ ( 𝜑 → Ⅎ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
21	20	19.9d	⊢ ( 𝜑 → ( ∃ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
22		opeq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
23		copsexgw	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜓 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
24	23	bicomd	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
25	24	eqcoms	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
26	22 25	syl	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
27	26	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
28	27 7	bitrd	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
29	28	ex	⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
30	2 21 29	bj-exlimd	⊢ ( 𝜑 → ( ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
31	1 16 30	bj-exlimd	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
32	12 31	biimtrrid	⊢ ( 𝜑 → ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
33	9 11 32	mp2and	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )

Metamath Proof Explorer

Theorem copsex2d

Proof