Theorem repizf 3984

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 3983. It is identical to zfrep6 3985 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 1990	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 2454	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)
3		ax-coll.1	. . 3 ⊢ Ⅎ𝑏𝜑
4	3	ax-coll 3983	. 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 1404  ∃wex 1436  ∃!weu 1960  ∀wral 2375  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-coll 3983

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-eu 1963  df-ral 2380

This theorem is referenced by:  zfrep6  3985  repizf2  4026