Theorem repizf 4200

Description: Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 4199. It is identical to zfrep6 4201 except for the choice of a freeness hypothesis rather than a disjoint variable condition between 𝑏 and 𝜑. (Contributed by Jim Kingdon, 23-Aug-2018.)

Hypothesis

Ref	Expression
ax-coll.1	⊢ Ⅎ𝑏𝜑

Assertion

Ref	Expression
repizf	⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)

Distinct variable group:   𝑥,𝑦,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem repizf

Step	Hyp	Ref	Expression
1		euex 2107	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 2593	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)
3		ax-coll.1	. . 3 ⊢ Ⅎ𝑏𝜑
4	3	ax-coll 4199	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)
5	2, 4	syl 14	1 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)

Syntax hints:   → wi 4  Ⅎwnf 1506  ∃wex 1538  ∃!weu 2077  ∀wral 2508  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-coll 4199

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-ral 2513

This theorem is referenced by:  zfrep6  4201  repizf2  4246