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Theorem bj-raldifsn 37301
Description: All elements in a set satisfy a given property if and only if all but one satisfy that property and that one also does. Typically, this can be used for characterizations that are proved using different methods for a given element and for all others, for instance zero and nonzero numbers, or the empty set and nonempty sets. (Contributed by BJ, 7-Dec-2021.)
Hypothesis
Ref Expression
bj-raldifsn.is (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
bj-raldifsn (𝐵𝐴 → (∀𝑥𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-raldifsn
StepHypRef Expression
1 difsnid 4766 . . . 4 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
21eqcomd 2742 . . 3 (𝐵𝐴𝐴 = ((𝐴 ∖ {𝐵}) ∪ {𝐵}))
32raleqdv 3296 . 2 (𝐵𝐴 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})𝜑))
4 ralunb 4149 . . 3 (∀𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
54a1i 11 . 2 (𝐵𝐴 → (∀𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)))
6 bj-raldifsn.is . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
76ralsng 4632 . . 3 (𝐵𝐴 → (∀𝑥 ∈ {𝐵}𝜑𝜓))
87anbi2d 630 . 2 (𝐵𝐴 → ((∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑𝜓)))
93, 5, 83bitrd 305 1 (𝐵𝐴 → (∀𝑥𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1541  wcel 2113  wral 3051  cdif 3898  cun 3899  {csn 4580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-sn 4581
This theorem is referenced by:  bj-0int  37302
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