Theorem ralunsn 4854

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 4143	. 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2		ralunsn.1	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	ralsng 4636	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜓))
4	3	anbi2d 642	. 2 ⊢ (𝐵 ∈ 𝐶 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
5	1, 4	bitrid 286	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-un 3904  df-sn 4585

This theorem is used by:  2ralunsn  4855  naddsuc2  8711  symgextfo  19636  gsmsymgrfixlem1  19641  gsmsymgreqlem2  19645  symgfixf1  19651  cply1coe0bi  22620  scmatf1  22846  mdetunilem9  22935  m2cpminvid2lem  23072  tgcgr4  28994  clwlkclwwlklem2a1  30583  clwlkclwwlkf1lem3  30597  disjunsn  33188