Theorem rmoi 3136

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	|- (x = B -> (ph <-> ps))
rmoi.c	|- (x = C -> (ph <-> ch))

Assertion

Ref	Expression
rmoi	|- ((E*x e. A ph /\ (B e. A /\ ps) /\ (C e. A /\ ch)) -> B = C)

Distinct variable groups:   x,A   x,B   x,C   ps,x   ch,x

Allowed substitution hint:   ph(x)

Proof of Theorem rmoi

Step	Hyp	Ref	Expression
1		rmoi.b	. . 3 |- (x = B -> (ph <-> ps))
2		rmoi.c	. . 3 |- (x = C -> (ph <-> ch))
3	1, 2	rmob 3135	. 2 |- ((E*x e. A ph /\ (B e. A /\ ps)) -> (B = C <-> (C e. A /\ ch)))
4	3	biimp3ar 1282	1 |- ((E*x e. A ph /\ (B e. A /\ ps) /\ (C e. A /\ ch)) -> B = C)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rmo 2623  df-v 2862

This theorem is referenced by: (None)