Theorem rexbidv2 3191

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
rexbidv2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
rexbidv2	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rexbidv2

Step	Hyp	Ref	Expression
1		rexbidv2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
2	1	exbidv 1948	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
3		df-rex 3096	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rex 3096	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-rex 3096

This theorem is referenced by:  rexbidva  3193  rexeqbidv  3346  rexssOLD  4021  iuneq12d  4990  exopxfr2  5831  isoini  7337  rexsupp  8178  omabs  8637  elfi2  9374  wemapsolem  9512  ltexpi  10887  rexuz  12922  ncoprmgcdne1b  16708  lpigen  21472  llyi  23600  nllyi  23601  elpi1  25173  ressupprn  32976  xrecex  33180  constrcbvlem  34090  bnj18eq1  35260  ldual1dim  39864  pmapjat1  40551  mrefg2  43364  islssfg2  43724  fourierdlem71  46817  hoiqssbl  47265  reuxfr1dd  49504  lubeldm2d  49655  glbeldm2d  49656