Theorem rexbii2 3105

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 1880	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
3		df-rex 3087	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3087	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812  df-rex 3087

This theorem is used by:  rexbiia  3107  rexeqbii  3333  rexrab  3653  rexin  4147  rexdifpr  4616  rexdifsn  4753  reusv2lem4  5362  reusv2  5364  frpoind  6345  eldifsucnn  8677  frind  9758  rexuz2  13030  rexrp  13147  rexuz3  15522  infpn2  17097  efgrelexlemb  19970  cmpcov2  23714  cmpfi  23732  txkgen  23977  cubic  27183  madeval2  28225  sumdmdii  33023  extdgfialglem1  34371  bnj882  35603  bnj893  35605  heibor1  38838  eldmqsres  39319  prtlem100  40010  islmodfg  44165  iuneq1i  46195  limcrecl  46735