Theorem eupick 1671

Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps regardless of x. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192.

Assertion

Ref	Expression
eupick	|- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Proof of Theorem eupick

Step	Hyp	Ref	Expression
1		mopick 1670	. 2 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2		eumo 1644	. 2 |- (E!xph -> E*xph)
3	1, 2	sylan 495	1 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 3   /\ wa 239  E.wex 1164  E!weu 1609  E*wmo 1610

This theorem is referenced by:  eupicka 1672  eupickb 1673  reupick 2700  copsexg 3352  funssres 4271  tz6.12-1 4504  chcmhi 10538  iotasbc 16065

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614

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