Theorem 2ralunsn 3881

Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Hypotheses

Ref	Expression
2ralunsn.1	⊢ (x = B → (φ ↔ χ))
2ralunsn.2	⊢ (y = B → (φ ↔ ψ))
2ralunsn.3	⊢ (x = B → (ψ ↔ θ))

Assertion

Ref	Expression
2ralunsn	⊢ (B ∈ C → (∀x ∈ (A ∪ {B})∀y ∈ (A ∪ {B})φ ↔ ((∀x ∈ A ∀y ∈ A φ ∧ ∀x ∈ A ψ) ∧ (∀y ∈ A χ ∧ θ))))

Distinct variable groups:   x,A   x,B,y   x,C   χ,x   ψ,y   θ,x

Allowed substitution hints:   φ(x,y)   ψ(x)   χ(y)   θ(y)   A(y)   C(y)

Proof of Theorem 2ralunsn

Step	Hyp	Ref	Expression
1		2ralunsn.2	. . . 4 ⊢ (y = B → (φ ↔ ψ))
2	1	ralunsn 3880	. . 3 ⊢ (B ∈ C → (∀y ∈ (A ∪ {B})φ ↔ (∀y ∈ A φ ∧ ψ)))
3	2	ralbidv 2635	. 2 ⊢ (B ∈ C → (∀x ∈ (A ∪ {B})∀y ∈ (A ∪ {B})φ ↔ ∀x ∈ (A ∪ {B})(∀y ∈ A φ ∧ ψ)))
4		2ralunsn.1	. . . . . 6 ⊢ (x = B → (φ ↔ χ))
5	4	ralbidv 2635	. . . . 5 ⊢ (x = B → (∀y ∈ A φ ↔ ∀y ∈ A χ))
6		2ralunsn.3	. . . . 5 ⊢ (x = B → (ψ ↔ θ))
7	5, 6	anbi12d 691	. . . 4 ⊢ (x = B → ((∀y ∈ A φ ∧ ψ) ↔ (∀y ∈ A χ ∧ θ)))
8	7	ralunsn 3880	. . 3 ⊢ (B ∈ C → (∀x ∈ (A ∪ {B})(∀y ∈ A φ ∧ ψ) ↔ (∀x ∈ A (∀y ∈ A φ ∧ ψ) ∧ (∀y ∈ A χ ∧ θ))))
9		r19.26 2747	. . . 4 ⊢ (∀x ∈ A (∀y ∈ A φ ∧ ψ) ↔ (∀x ∈ A ∀y ∈ A φ ∧ ∀x ∈ A ψ))
10	9	anbi1i 676	. . 3 ⊢ ((∀x ∈ A (∀y ∈ A φ ∧ ψ) ∧ (∀y ∈ A χ ∧ θ)) ↔ ((∀x ∈ A ∀y ∈ A φ ∧ ∀x ∈ A ψ) ∧ (∀y ∈ A χ ∧ θ)))
11	8, 10	syl6bb 252	. 2 ⊢ (B ∈ C → (∀x ∈ (A ∪ {B})(∀y ∈ A φ ∧ ψ) ↔ ((∀x ∈ A ∀y ∈ A φ ∧ ∀x ∈ A ψ) ∧ (∀y ∈ A χ ∧ θ))))
12	3, 11	bitrd 244	1 ⊢ (B ∈ C → (∀x ∈ (A ∪ {B})∀y ∈ (A ∪ {B})φ ↔ ((∀x ∈ A ∀y ∈ A φ ∧ ∀x ∈ A ψ) ∧ (∀y ∈ A χ ∧ θ))))

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: (None)