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Axiom ax-ie2 1292
Description: Define existential quantification. xφ means "there exists at least one set x such that φ is true." Axiom 10 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ax-ie2 (x(ψxψ) → (x(φψ) ↔ (xφψ)))

Detailed syntax breakdown of Axiom ax-ie2
StepHypRef Expression
1 wps . . . 4 wff ψ
2 vx . . . . 5 set x
31, 2wal 1251 . . . 4 wff xψ
41, 3wi 4 . . 3 wff (ψxψ)
54, 2wal 1251 . 2 wff x(ψxψ)
6 wph . . . . 5 wff φ
76, 1wi 4 . . . 4 wff (φψ)
87, 2wal 1251 . . 3 wff x(φψ)
96, 2wex 1290 . . . 4 wff xφ
109, 1wi 4 . . 3 wff (xφψ)
118, 10wb 97 . 2 wff (x(φψ) ↔ (xφψ))
125, 11wi 4 1 wff (x(ψxψ) → (x(φψ) ↔ (xφψ)))
Colors of variables: wff set class
This axiom is referenced by:  19.23t  1294
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