Theorem repizf 4205

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 4204. It is identical to zfrep6 4206 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 2109	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 2595	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)
3		ax-coll.1	. . 3 ⊢ Ⅎ𝑏𝜑
4	3	ax-coll 4204	. 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 1508  ∃wex 1540  ∃!weu 2079  ∀wral 2510  ∃wrex 2511

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-coll 4204

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-ral 2515

This theorem is referenced by:  zfrep6  4206  repizf2  4252