Theorem ralunsn 4837

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 4137	. 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2		ralunsn.1	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	ralsng 4620	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜓))
4	3	anbi2d 629	. 2 ⊢ (𝐵 ∈ 𝐶 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
5	1, 4	bitrid 282	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ∪ cun 3895  {csn 4572

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3443  df-un 3902  df-sn 4573

This theorem is referenced by:  2ralunsn  4838  symgextfo  19118  gsmsymgrfixlem1  19123  gsmsymgreqlem2  19127  symgfixf1  19133  cply1coe0bi  21569  scmatf1  21778  mdetunilem9  21867  m2cpminvid2lem  22001  tgcgr4  27122  clwlkclwwlklem2a1  28585  clwlkclwwlkf1lem3  28599  disjunsn  31161