Theorem ralunsn 4852

Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
ralunsn.1	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralunsn	⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Distinct variable groups:   𝑥,𝐵   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem ralunsn

Step	Hyp	Ref	Expression
1		ralunb 4149	. 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2		ralunsn.1	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	ralsng 4634	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜓))
4	3	anbi2d 639	. 2 ⊢ (𝐵 ∈ 𝐶 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
5	1, 4	bitrid 285	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∪ cun 3902  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-v 3456  df-un 3909  df-sn 4583

This theorem is referenced by:  2ralunsn  4853  naddsuc2  8672  symgextfo  19462  gsmsymgrfixlem1  19467  gsmsymgreqlem2  19471  symgfixf1  19477  cply1coe0bi  22365  scmatf1  22591  mdetunilem9  22680  m2cpminvid2lem  22814  tgcgr4  28700  clwlkclwwlklem2a1  30194  clwlkclwwlkf1lem3  30208  disjunsn  32794