Theorem rexbidv2 3181

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 1920	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
3		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rex 3077	. 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 1777   ∈ wcel 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-rex 3077

This theorem is referenced by:  rexbidva  3183  rexeqbidv  3355  rexss  4084  iuneq12d  5044  exopxfr2  5869  isoini  7374  rexsupp  8223  omabs  8707  elfi2  9483  wemapsolem  9619  ltexpi  10971  rexuz  12963  ncoprmgcdne1b  16697  lpigen  21368  llyi  23503  nllyi  23504  elpi1  25097  ressupprn  32702  xrecex  32884  bnj18eq1  34903  ldual1dim  39122  pmapjat1  39810  mrefg2  42663  islssfg2  43028  fourierdlem71  46098  hoiqssbl  46546  lubeldm2d  48638  glbeldm2d  48639