Theorem 2rexbiia 3206

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Hypothesis

Ref	Expression
2rexbiia.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
2rexbiia	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓))
2	1	rexbidva 3170	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
3	2	rexbiia 3092	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2104  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1780  df-rex 3072

This theorem is referenced by:  reu3op  6210  opreu2reurex  6212  cnref1o  12775  mndpfo  18457  wspthsnwspthsnon  28330  mdsymlem8  30821  xlt2addrd  31130  elunirnmbfm  32269  satfv0  33369  fmla0xp  33394  icoreelrnab  35573  rrx2xpref1o  46308