Theorem ralprg 4658

Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2178, ax-12 2215. (Revised by GG, 30-Sep-2024.)

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 4588	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2	1	raleqi 3321	. . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3		ralunb 4152	. . 3 ⊢ (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
4	2, 3	bitri 278	. 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5		ralprg.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	ralsng 4637	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
7		ralprg.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
8	7	ralsng 4637	. . 3 ⊢ (𝐵 ∈ 𝑊 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜒))
9	6, 8	bi2anan9 649	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∧ 𝜒)))
10	4, 9	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ∪ cun 3905  {csn 4585  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-v 3459  df-un 3912  df-sn 4586  df-pr 4588

This theorem is referenced by:  rexprg  4659  raltpg  4660  ralpr  4662  reuprg0  4664  iinxprg  5051  disjprg  5101  fpropnf1  7255  f12dfv  7261  f13dfv  7262  suppr  9420  infpr  9453  pfx2  14974  sumpr  15789  gcdcllem2  16548  lcmfpr  16675  joinval2lem  18424  meetval2lem  18438  sgrp2rid2  18978  sgrp2nmndlem4  18980  sgrp2nmndlem5  18981  iccntr  24940  limcun  26015  cplgr3v  29694  3wlkdlem4  30422  frgr3v  30535  3vfriswmgr  30538  prsiga  34438  paireqne  48115