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Theorem spimeh 1667
 Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
Hypotheses
Ref Expression
spimeh.1 (φxφ)
spimeh.2 (x = z → (φψ))
Assertion
Ref Expression
spimeh (φxψ)
Distinct variable group:   x,z
Allowed substitution hints:   φ(x,z)   ψ(x,z)

Proof of Theorem spimeh
StepHypRef Expression
1 spimeh.1 . 2 (φxφ)
2 a9ev 1656 . . . 4 x x = z
3 spimeh.2 . . . . 5 (x = z → (φψ))
43eximi 1576 . . . 4 (x x = zx(φψ))
52, 4ax-mp 5 . . 3 x(φψ)
6519.35i 1601 . 2 (xφxψ)
71, 6syl 15 1 (φxψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541 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-9 1654 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  ax12olem1  1927  ax10lem2  1937
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