Theorem zfrepclf 5218

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
zfrepclf.1	⊢ Ⅎ𝑥𝐴
zfrepclf.2	⊢ 𝐴 ∈ V
zfrepclf.3	⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem zfrepclf

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 ⊢ 𝐴 ∈ V
2		zfrepclf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2924	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = 𝐴
4		eleq2 2827	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ 𝐴))
5		zfrepclf.3	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
6	4, 5	syl6bi 252	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
7	3, 6	alrimi 2206	. . . 4 ⊢ (𝑣 = 𝐴 → ∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
8		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧𝜑
9	8	axrep5 5215	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)))
10	7, 9	syl 17	. . 3 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)))
11	4	anbi1d 630	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑥 ∈ 𝑣 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
12	3, 11	exbid 2216	. . . . . 6 ⊢ (𝑣 = 𝐴 → (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
13	12	bibi2d 343	. . . . 5 ⊢ (𝑣 = 𝐴 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
14	13	albidv 1923	. . . 4 ⊢ (𝑣 = 𝐴 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
15	14	exbidv 1924	. . 3 ⊢ (𝑣 = 𝐴 → (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
16	10, 15	mpbid 231	. 2 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
17	1, 16	vtocle 3524	1 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Ⅎwnfc 2887  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889

This theorem is referenced by:  zfrep3cl  5219  zfrep4  5220