Theorem opelopabg 2115

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61.

Hypotheses

Ref	Expression
opelopabg.1	⊢ (x = A → (φ ↔ ψ))
opelopabg.2	⊢ (y = B → (ψ ↔ χ))

Assertion

Ref	Expression
opelopabg	⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))

Distinct variable group(s):   x,y,A   x,B,y   χ,x,y

Proof of Theorem opelopabg

Step	Hyp	Ref	Expression
1		elex 1356	. . . 4 ⊢ (A ∈ C → ∃x x = A)
2		elex 1356	. . . 4 ⊢ (B ∈ D → ∃y y = B)
3	1, 2	anim12i 268	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x x = A ∧ ∃y y = B))
4		eeanv 980	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
5	3, 4	sylibr 175	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → ∃x∃y(x = A ∧ y = B))
6		ax-17 925	. . . . 5 ⊢ (z ∈ ⟨A, B⟩ → ∀x z ∈ ⟨A, B⟩)
7		hbopab1 2112	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀x z ∈ {⟨x, y⟩∣φ})
8	6, 7	hbel 1172	. . . 4 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} → ∀x⟨A, B⟩ ∈ {⟨x, y⟩∣φ})
9		ax-17 925	. . . 4 ⊢ (χ → ∀xχ)
10	8, 9	hbbi 705	. . 3 ⊢ ((⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ) → ∀x(⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
11		ax-17 925	. . . . . 6 ⊢ (z ∈ ⟨A, B⟩ → ∀y z ∈ ⟨A, B⟩)
12		hbopab2 2113	. . . . . 6 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})
13	11, 12	hbel 1172	. . . . 5 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} → ∀y⟨A, B⟩ ∈ {⟨x, y⟩∣φ})
14		ax-17 925	. . . . 5 ⊢ (χ → ∀yχ)
15	13, 14	hbbi 705	. . . 4 ⊢ ((⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ) → ∀y(⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
16		opeq12 1878	. . . . . . 7 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
17	16	eleq1d 1155	. . . . . 6 ⊢ ((x = A ∧ y = B) → (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣φ}))
18		opabid 2099	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)
19	17, 18	syl5bbr 412	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣φ}))
20		opelopabg.1	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
21		opelopabg.2	. . . . . 6 ⊢ (y = B → (ψ ↔ χ))
22	20, 21	sylan9bb 418	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ χ))
23	19, 22	bitr3d 408	. . . 4 ⊢ ((x = A ∧ y = B) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
24	15, 23	19.23ai 746	. . 3 ⊢ (∃y(x = A ∧ y = B) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
25	10, 24	19.23ai 746	. 2 ⊢ (∃x∃y(x = A ∧ y = B) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
26	5, 25	syl 12	1 ⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055

This theorem is referenced by:  brabg 2116  opelopab 2117  opelcnvg 2517  fvopab3 2868  fvopab3ig 2869  fvopabn 2873  oprabval 3047  brecop 3242

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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