Theorem repizf 3914

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 3913. It is identical to zfrep6 3915 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 1973	. . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 2431	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)
3		ax-coll.1	. . 3 ⊢ Ⅎ𝑏𝜑
4	3	ax-coll 3913	. 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 1390  ∃wex 1422  ∃!weu 1943  ∀wral 2353  ∃wrex 2354

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-coll 3913

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-ral 2358

This theorem is referenced by:  zfrep6  3915  repizf2  3956