Theorem rexbii2 3096

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
rexbii2.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))

Assertion

Ref	Expression
rexbii2	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Proof of Theorem rexbii2

Step	Hyp	Ref	Expression
1		rexbii2.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
2	1	exbii 1846	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
3		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-rex 3077

This theorem is referenced by:  rexbiia  3098  rexeqbii  3353  rexrab  3718  rexin  4269  rexdifpr  4681  rexdifsn  4819  reusv2lem4  5419  reusv2  5421  frpoind  6374  wfiOLD  6383  eldifsucnn  8720  frind  9819  rexuz2  12964  rexrp  13078  rexuz3  15397  infpn2  16960  efgrelexlemb  19792  cmpcov2  23419  cmpfi  23437  txkgen  23681  cubic  26910  madeval2  27910  sumdmdii  32447  bnj882  34902  bnj893  34904  heibor1  37770  eldmqsres  38243  prtlem100  38815  islmodfg  43026  iuneq1i  44987  limcrecl  45550