Theorem rexbidva 2317

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)

Hypothesis

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

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem rexbidva

Step	Hyp	Ref	Expression
1		nfv 1418	. 2 ⊢ Ⅎxφ
2		ralbidva.1	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	1, 2	rexbida 2315	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-rex 2306

This theorem is referenced by:  2rexbiia  2334  2rexbidva  2341  rexeqbidva  2514  dfimafn  5165  funimass4  5167  fconstfvm  5322  fliftel  5376  fliftf  5382  f1oiso  5408  releldm2  5753  qsinxp  6118  qliftel  6122  genpassl  6506  genpassu  6507  addcomprg  6553  mulcomprg  6555  1idprl  6565  1idpru  6566  archsr  6688  cnegexlem3  6965  cnegex2  6967  recexre  7342  rerecclap  7468  creur  7672  creui  7673  nndiv  7715  arch  7934  nnrecl  7935  expnlbnd  9006