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Theorem rexbii2 2643
Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
rexbii2.1 ((x A φ) ↔ (x B ψ))
Assertion
Ref Expression
rexbii2 (x A φx B ψ)

Proof of Theorem rexbii2
StepHypRef Expression
1 rexbii2.1 . . 3 ((x A φ) ↔ (x B ψ))
21exbii 1582 . 2 (x(x A φ) ↔ x(x B ψ))
3 df-rex 2620 . 2 (x A φx(x A φ))
4 df-rex 2620 . 2 (x B ψx(x B ψ))
52, 3, 43bitr4i 268 1 (x A φx B ψ)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   wcel 1710  wrex 2615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557
This theorem depends on definitions:  df-bi 177  df-ex 1542  df-rex 2620
This theorem is referenced by:  rexeqbii  2645  rexbiia  2647  rexrab  3000  rexdifsn  3843  pw1in  4164
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