Theorem rmobiia 2802

Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmobiia.1	|- (x e. A -> (ph <-> ps))

Assertion

Ref	Expression
rmobiia	|- (E*x e. A ph <-> E*x e. A ps)

Proof of Theorem rmobiia

Step	Hyp	Ref	Expression
1		rmobiia.1	. . . 4 |- (x e. A -> (ph <-> ps))
2	1	pm5.32i 618	. . 3 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	mobii 2240	. 2 |- (E*x(x e. A /\ ph) <-> E*x(x e. A /\ ps))
4		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
5		df-rmo 2623	. 2 |- (E*x e. A ps <-> E*x(x e. A /\ ps))
6	3, 4, 5	3bitr4i 268	1 |- (E*x e. A ph <-> E*x e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  E*wmo 2205  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208  df-mo 2209  df-rmo 2623

This theorem is referenced by:  rmobii  2803