Theorem ralpr 3780

Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralpr.1	⊢ A ∈ V
ralpr.2	⊢ B ∈ V
ralpr.3	⊢ (x = A → (φ ↔ ψ))
ralpr.4	⊢ (x = B → (φ ↔ χ))

Assertion

Ref	Expression
ralpr	⊢ (∀x ∈ {A, B}φ ↔ (ψ ∧ χ))

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

Allowed substitution hint:   φ(x)

Proof of Theorem ralpr

Step	Hyp	Ref	Expression
1		ralpr.1	. 2 ⊢ A ∈ V
2		ralpr.2	. 2 ⊢ B ∈ V
3		ralpr.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
4		ralpr.4	. . 3 ⊢ (x = B → (φ ↔ χ))
5	3, 4	ralprg 3776	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (∀x ∈ {A, B}φ ↔ (ψ ∧ χ)))
6	1, 2, 5	mp2an 653	1 ⊢ (∀x ∈ {A, B}φ ↔ (ψ ∧ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860  {cpr 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 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

This theorem is referenced by: (None)