Theorem ralpr 4656

Description: Convert a restricted universal 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 4652	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
6	1, 2, 5	mp2an 693	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  Vcvv 3439  {cpr 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-v 3441  df-un 3905  df-sn 4580  df-pr 4582

This theorem is referenced by:  fprb  7140  fzprval  13503  fvinim0ffz  13707  wwlktovf1  14882  xpsfrnel  17485  xpsle  17502  isdrs2  18231  pmtrsn  19450  iblcnlem1  25747  lfuhgr1v0e  29308  nbgr2vtx1edg  29404  nbuhgr2vtx1edgb  29406  umgr2v2evd2  29582  2wlklem  29720  dfpth2  29783  2wlkdlem5  29983  2wlkdlem10  29989  clwwlknonex2lem2  30164  3pthdlem1  30220  upgr4cycl4dv4e  30241  subfacp1lem3  35355  poimirlem1  37791  paireqne  47794  requad2  47906  ldepsnlinc  48791  rrx2pnecoorneor  48998  rrx2line  49023  rrx2linest  49025