Theorem rexbii2 3108

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 1878	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
3		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-ex 1810  df-rex 3090

This theorem is referenced by:  rexbiia  3110  rexeqbii  3337  rexrab  3660  rexin  4204  rexdifpr  4626  rexdifsn  4763  reusv2lem4  5374  reusv2  5376  frpoind  6345  eldifsucnn  8651  frind  9723  rexuz2  12924  rexrp  13040  rexuz3  15402  infpn2  16974  efgrelexlemb  19821  cmpcov2  23528  cmpfi  23546  txkgen  23790  cubic  26995  madeval2  28007  sumdmdii  32748  extdgfialglem1  34063  bnj882  35295  bnj893  35297  heibor1  38442  eldmqsres  38923  prtlem100  39614  islmodfg  43779  iuneq1i  45787  limcrecl  46328