Theorem rexcom4b 2881

Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Hypothesis

Ref	Expression
rexcom4b.1	⊢ B ∈ V

Assertion

Ref	Expression
rexcom4b	⊢ (∃x∃y ∈ A (φ ∧ x = B) ↔ ∃y ∈ A φ)

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

Allowed substitution hints:   φ(y)   A(y)   B(y)

Proof of Theorem rexcom4b

Step	Hyp	Ref	Expression
1		rexcom4a 2880	. 2 ⊢ (∃x∃y ∈ A (φ ∧ x = B) ↔ ∃y ∈ A (φ ∧ ∃x x = B))
2		rexcom4b.1	. . . . 5 ⊢ B ∈ V
3	2	isseti 2866	. . . 4 ⊢ ∃x x = B
4	3	biantru 491	. . 3 ⊢ (φ ↔ (φ ∧ ∃x x = B))
5	4	rexbii 2640	. 2 ⊢ (∃y ∈ A φ ↔ ∃y ∈ A (φ ∧ ∃x x = B))
6	1, 5	bitr4i 243	1 ⊢ (∃x∃y ∈ A (φ ∧ x = B) ↔ ∃y ∈ A φ)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860

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-rex 2621  df-v 2862

This theorem is referenced by: (None)