Theorem repizf 3847

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 3846. It is identical to zfrep6 3848 except for the choice of a freeness hypothesis rather than a distinct variable constraint 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 1912	. . 3 ⊢ (∃!yφ → ∃yφ)
2	1	ralimi 2362	. 2 ⊢ (∀x ∈ 𝑎 ∃!yφ → ∀x ∈ 𝑎 ∃yφ)
3		ax-coll.1	. . 3 ⊢ Ⅎ𝑏φ
4	3	ax-coll 3846	. 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 1329  ∃wex 1362  ∃!weu 1882  ∀wral 2284  ∃wrex 2285

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-coll 3846

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-ral 2289

This theorem is referenced by:  zfrep6  3848  repizf2  3889