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Theorem ralprg 3449
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 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
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 3410 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21raleqi 2526 . . 3 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 ralunb 3152 . . 3 (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
42, 3bitri 177 . 2 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5 ralprg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65ralsng 3439 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
7 ralprg.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
87ralsng 3439 . . 3 (𝐵𝑊 → (∀𝑥 ∈ {𝐵}𝜑𝜒))
96, 8bi2anan9 548 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
104, 9syl5bb 185 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  cun 2943  {csn 3403  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2788  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by:  raltpg  3451  ralpr  3453  iinxprg  3759  fvinim0ffz  9198
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