Theorem rexbidv2 3185

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 1951	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
3		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rex 3090	. 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 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  ax-5 1940

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

This theorem is referenced by:  rexbidva  3187  rexeqbidv  3339  rexssOLD  4013  iuneq12d  4986  exopxfr2  5830  isoini  7336  rexsupp  8174  omabs  8633  elfi2  9370  wemapsolem  9508  ltexpi  10882  rexuz  12917  ncoprmgcdne1b  16703  lpigen  21503  llyi  23631  nllyi  23632  elpi1  25204  ressupprn  33035  xrecex  33239  constrcbvlem  34145  bnj18eq1  35315  ldual1dim  39940  pmapjat1  40627  mrefg2  43438  islssfg2  43798  fourierdlem71  46891  hoiqssbl  47339  reuxfr1dd  49585  lubeldm2d  49736  glbeldm2d  49737