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Axiom ax-pr 3999
Description: The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3925). (Contributed by NM, 14-Nov-2006.)
Assertion
Ref Expression
ax-pr 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-pr
StepHypRef Expression
1 vw . . . . . 6 setvar 𝑤
2 vx . . . . . 6 setvar 𝑥
31, 2weq 1433 . . . . 5 wff 𝑤 = 𝑥
4 vy . . . . . 6 setvar 𝑦
51, 4weq 1433 . . . . 5 wff 𝑤 = 𝑦
63, 5wo 662 . . . 4 wff (𝑤 = 𝑥𝑤 = 𝑦)
7 vz . . . . 5 setvar 𝑧
81, 7wel 1435 . . . 4 wff 𝑤𝑧
96, 8wi 4 . . 3 wff ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
109, 1wal 1283 . 2 wff 𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
1110, 7wex 1422 1 wff 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff set class
This axiom is referenced by:  zfpair2  4000  bj-zfpair2  11142
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