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  6507  genpassu  6508  addcomprg  6554  mulcomprg  6556  1idprl  6566  1idpru  6567  archrecnq  6635  archsr  6708  cnegexlem3  6985  cnegex2  6987  recexre  7362  rerecclap  7488  creur  7692  creui  7693  nndiv  7735  arch  7954  nnrecl  7955  expnlbnd  9026