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Theorem rexbidv2 2323
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 ψx B χ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)   B(x)

Proof of Theorem rexbidv2
StepHypRef Expression
1 rexbidv2.1 . . 3 (φ → ((x A ψ) ↔ (x B χ)))
21exbidv 1703 . 2 (φ → (x(x A ψ) ↔ x(x B χ)))
3 df-rex 2306 . 2 (x A ψx(x A ψ))
4 df-rex 2306 . 2 (x B χx(x B χ))
52, 3, 43bitr4g 212 1 (φ → (x A ψx B χ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ 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-rex 2306 This theorem is referenced by:  rexss  3001  rexsupp  5232  isoini  5398  ltexpi  6314  rexuz  8248
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