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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 → (φψ))
ralprg.2 (x = B → (φχ))
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
StepHypRef Expression
1 df-pr 3374 . . . 4 {A, B} = ({A} ∪ {B})
21raleqi 2503 . . 3 (x {A, B}φx ({A} ∪ {B})φ)
3 ralunb 3118 . . 3 (x ({A} ∪ {B})φ ↔ (x {A}φ x {B}φ))
42, 3bitri 173 . 2 (x {A, B}φ ↔ (x {A}φ x {B}φ))
5 ralprg.1 . . . 4 (x = A → (φψ))
65ralsng 3402 . . 3 (A 𝑉 → (x {A}φψ))
7 ralprg.2 . . . 4 (x = B → (φχ))
87ralsng 3402 . . 3 (B 𝑊 → (x {B}φχ))
96, 8bi2anan9 538 . 2 ((A 𝑉 B 𝑊) → ((x {A}φ x {B}φ) ↔ (ψ χ)))
104, 9syl5bb 181 1 ((A 𝑉 B 𝑊) → (x {A, B}φ ↔ (ψ χ)))
 Colors of variables: wff set class 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
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