Theorem ralpr 4238

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 4234	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
6	1, 2, 5	mp2an 708	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  fzprval  12401  fvinim0ffz  12587  wwlktovf1  13700  xpsfrnel  16223  xpsle  16241  isdrs2  16939  pmtrsn  17939  iblcnlem1  23554  lfuhgr1v0e  26146  nbgr2vtx1edg  26246  nbuhgr2vtx1edgb  26248  umgr2v2evd2  26423  2wlklem  26563  2wlkdlem5  26825  2wlkdlem10  26831  3pthdlem1  27024  upgr4cycl4dv4e  27045  numclwwlkovf2exlem2  27212  subfacp1lem3  31164  fprb  31669  poimirlem1  33410  ldepsnlinc  42297