Theorem rexbidv2 3154

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 1921	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
3		df-rex 3055	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rex 3055	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054

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 1910

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-rex 3055

This theorem is referenced by:  rexbidva  3156  rexeqbidv  3322  rexssOLD  4027  iuneq12d  4988  exopxfr2  5811  isoini  7316  rexsupp  8164  omabs  8618  elfi2  9372  wemapsolem  9510  ltexpi  10862  rexuz  12864  ncoprmgcdne1b  16627  lpigen  21252  llyi  23368  nllyi  23369  elpi1  24952  ressupprn  32620  xrecex  32847  constrcbvlem  33752  bnj18eq1  34924  ldual1dim  39166  pmapjat1  39854  mrefg2  42702  islssfg2  43067  fourierdlem71  46182  hoiqssbl  46630  reuxfr1dd  48799  lubeldm2d  48950  glbeldm2d  48951