Theorem zfrep3cl 5257

Description: An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)

Hypotheses

Ref	Expression
zfrep3cl.1	⊢ 𝐴 ∈ V
zfrep3cl.2	⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))

Assertion

Ref	Expression
zfrep3cl	⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem zfrep3cl

Step	Hyp	Ref	Expression
1		nfcv 2931	. 2 ⊢ Ⅎ𝑥𝐴
2		zfrep3cl.1	. 2 ⊢ 𝐴 ∈ V
3		zfrep3cl.2	. 2 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
4	1, 2, 3	zfrepclf 5256	1 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149  Vcvv 3463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-cleq 2761  df-clel 2844  df-nfc 2918

This theorem is referenced by: (None)