Theorem raltpg 3777

Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralprg.1	⊢ (x = A → (φ ↔ ψ))
ralprg.2	⊢ (x = B → (φ ↔ χ))
raltpg.3	⊢ (x = C → (φ ↔ θ))

Assertion

Ref	Expression
raltpg	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ)))

Distinct variable groups:   x,A   x,B   x,C   ψ,x   χ,x   θ,x

Allowed substitution hints:   φ(x)   V(x)   W(x)   X(x)

Proof of Theorem raltpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
2		ralprg.2	. . . . 5 ⊢ (x = B → (φ ↔ χ))
3	1, 2	ralprg 3775	. . . 4 ⊢ ((A ∈ V ∧ B ∈ W) → (∀x ∈ {A, B}φ ↔ (ψ ∧ χ)))
4		raltpg.3	. . . . 5 ⊢ (x = C → (φ ↔ θ))
5	4	ralsng 3765	. . . 4 ⊢ (C ∈ X → (∀x ∈ {C}φ ↔ θ))
6	3, 5	bi2anan9 843	. . 3 ⊢ (((A ∈ V ∧ B ∈ W) ∧ C ∈ X) → ((∀x ∈ {A, B}φ ∧ ∀x ∈ {C}φ) ↔ ((ψ ∧ χ) ∧ θ)))
7	6	3impa 1146	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → ((∀x ∈ {A, B}φ ∧ ∀x ∈ {C}φ) ↔ ((ψ ∧ χ) ∧ θ)))
8		df-tp 3743	. . . 4 ⊢ {A, B, C} = ({A, B} ∪ {C})
9	8	raleqi 2811	. . 3 ⊢ (∀x ∈ {A, B, C}φ ↔ ∀x ∈ ({A, B} ∪ {C})φ)
10		ralunb 3444	. . 3 ⊢ (∀x ∈ ({A, B} ∪ {C})φ ↔ (∀x ∈ {A, B}φ ∧ ∀x ∈ {C}φ))
11	9, 10	bitri 240	. 2 ⊢ (∀x ∈ {A, B, C}φ ↔ (∀x ∈ {A, B}φ ∧ ∀x ∈ {C}φ))
12		df-3an 936	. 2 ⊢ ((ψ ∧ χ ∧ θ) ↔ ((ψ ∧ χ) ∧ θ))
13	7, 11, 12	3bitr4g 279	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by:  raltp  3781