Theorem r19.12sn 3790

Description: Special case of r19.12 2728 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
r19.12sn.1	⊢ A ∈ V

Assertion

Ref	Expression
r19.12sn	⊢ (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ)

Distinct variable groups:   x,y,A   x,B

Allowed substitution hints:   φ(x,y)   B(y)

Proof of Theorem r19.12sn

Step	Hyp	Ref	Expression
1		r19.12sn.1	. 2 ⊢ A ∈ V
2		sbcralg 3121	. . 3 ⊢ (A ∈ V → ([̣A / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ))
3		rexsns 3765	. . 3 ⊢ (A ∈ V → (∃x ∈ {A}∀y ∈ B φ ↔ [̣A / x]̣∀y ∈ B φ))
4		rexsns 3765	. . . 4 ⊢ (A ∈ V → (∃x ∈ {A}φ ↔ [̣A / x]̣φ))
5	4	ralbidv 2635	. . 3 ⊢ (A ∈ V → (∀y ∈ B ∃x ∈ {A}φ ↔ ∀y ∈ B [̣A / x]̣φ))
6	2, 3, 5	3bitr4d 276	. 2 ⊢ (A ∈ V → (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ))
7	1, 6	ax-mp 5	1 ⊢ (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860  [̣wsbc 3047  {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-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-rex 2621  df-v 2862  df-sbc 3048  df-sn 3742

This theorem is referenced by: (None)