Theorem ralunsn 3880

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 3445	. 2 ⊢ (∀x ∈ (A ∪ {B})φ ↔ (∀x ∈ A φ ∧ ∀x ∈ {B}φ))
2		ralunsn.1	. . . 4 ⊢ (x = B → (φ ↔ ψ))
3	2	ralsng 3766	. . 3 ⊢ (B ∈ C → (∀x ∈ {B}φ ↔ ψ))
4	3	anbi2d 684	. 2 ⊢ (B ∈ C → ((∀x ∈ A φ ∧ ∀x ∈ {B}φ) ↔ (∀x ∈ A φ ∧ ψ)))
5	1, 4	syl5bb 248	1 ⊢ (B ∈ C → (∀x ∈ (A ∪ {B})φ ↔ (∀x ∈ A φ ∧ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ∪ cun 3208  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742

This theorem is referenced by:  2ralunsn  3881