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Axiom ax-ie2 1383
Description: Define existential quantification. 𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ax-ie2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Detailed syntax breakdown of Axiom ax-ie2
StepHypRef Expression
1 wps . . . 4 wff 𝜓
2 vx . . . . 5 setvar 𝑥
31, 2wal 1241 . . . 4 wff 𝑥𝜓
41, 3wi 4 . . 3 wff (𝜓 → ∀𝑥𝜓)
54, 2wal 1241 . 2 wff 𝑥(𝜓 → ∀𝑥𝜓)
6 wph . . . . 5 wff 𝜑
76, 1wi 4 . . . 4 wff (𝜑𝜓)
87, 2wal 1241 . . 3 wff 𝑥(𝜑𝜓)
96, 2wex 1381 . . . 4 wff 𝑥𝜑
109, 1wi 4 . . 3 wff (∃𝑥𝜑𝜓)
118, 10wb 98 . 2 wff (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
125, 11wi 4 1 wff (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff set class
This axiom is referenced by:  19.23ht  1386  bj-ex  9751
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