Theorem rexcom4a 2880

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

Assertion

Ref	Expression
rexcom4a	⊢ (∃x∃y ∈ A (φ ∧ ψ) ↔ ∃y ∈ A (φ ∧ ∃xψ))

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

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

Proof of Theorem rexcom4a

Step	Hyp	Ref	Expression
1		rexcom4 2879	. 2 ⊢ (∃y ∈ A ∃x(φ ∧ ψ) ↔ ∃x∃y ∈ A (φ ∧ ψ))
2		19.42v 1905	. . 3 ⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))
3	2	rexbii 2640	. 2 ⊢ (∃y ∈ A ∃x(φ ∧ ψ) ↔ ∃y ∈ A (φ ∧ ∃xψ))
4	1, 3	bitr3i 242	1 ⊢ (∃x∃y ∈ A (φ ∧ ψ) ↔ ∃y ∈ A (φ ∧ ∃xψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  ∃wrex 2616

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:  rexcom4b  2881