Theorem raltp 3782

Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
raltp.1	⊢ A ∈ V
raltp.2	⊢ B ∈ V
raltp.3	⊢ C ∈ V
raltp.4	⊢ (x = A → (φ ↔ ψ))
raltp.5	⊢ (x = B → (φ ↔ χ))
raltp.6	⊢ (x = C → (φ ↔ θ))

Assertion

Ref	Expression
raltp	⊢ (∀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 hint:   φ(x)

Proof of Theorem raltp

Step	Hyp	Ref	Expression
1		raltp.1	. 2 ⊢ A ∈ V
2		raltp.2	. 2 ⊢ B ∈ V
3		raltp.3	. 2 ⊢ C ∈ V
4		raltp.4	. . 3 ⊢ (x = A → (φ ↔ ψ))
5		raltp.5	. . 3 ⊢ (x = B → (φ ↔ χ))
6		raltp.6	. . 3 ⊢ (x = C → (φ ↔ θ))
7	4, 5, 6	raltpg 3778	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ)))
8	1, 2, 3, 7	mp3an 1277	1 ⊢ (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860  {ctp 3740

This proof depends on 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 proof 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 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is used by: (None)