Theorem ralprg 3412

Description: Convert a quantification over a pair 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))

Assertion

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

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

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

Proof of Theorem ralprg

Step	Hyp	Ref	Expression
1		df-pr 3374	. . . 4 ⊢ {A, B} = ({A} ∪ {B})
2	1	raleqi 2503	. . 3 ⊢ (∀x ∈ {A, B}φ ↔ ∀x ∈ ({A} ∪ {B})φ)
3		ralunb 3118	. . 3 ⊢ (∀x ∈ ({A} ∪ {B})φ ↔ (∀x ∈ {A}φ ∧ ∀x ∈ {B}φ))
4	2, 3	bitri 173	. 2 ⊢ (∀x ∈ {A, B}φ ↔ (∀x ∈ {A}φ ∧ ∀x ∈ {B}φ))
5		ralprg.1	. . . 4 ⊢ (x = A → (φ ↔ ψ))
6	5	ralsng 3402	. . 3 ⊢ (A ∈ 𝑉 → (∀x ∈ {A}φ ↔ ψ))
7		ralprg.2	. . . 4 ⊢ (x = B → (φ ↔ χ))
8	7	ralsng 3402	. . 3 ⊢ (B ∈ 𝑊 → (∀x ∈ {B}φ ↔ χ))
9	6, 8	bi2anan9 538	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((∀x ∈ {A}φ ∧ ∀x ∈ {B}φ) ↔ (ψ ∧ χ)))
10	4, 9	syl5bb 181	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∀x ∈ {A, B}φ ↔ (ψ ∧ χ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∪ cun 2909  {csn 3367  {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  raltpg  3414  ralpr  3416  iinxprg  3722