Theorem zfrepclf 5229

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 2912	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = 𝐴
4		eleq2 2820	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ 𝐴))
5		zfrepclf.3	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
6	4, 5	biimtrdi 253	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
7	3, 6	alrimi 2216	. . . 4 ⊢ (𝑣 = 𝐴 → ∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
8		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧𝜑
9	8	axrep5 5225	. . . 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 631	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑥 ∈ 𝑣 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
12	3, 11	exbid 2226	. . . . . 6 ⊢ (𝑣 = 𝐴 → (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
13	12	bibi2d 342	. . . . 5 ⊢ (𝑣 = 𝐴 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
14	13	albidv 1921	. . . 4 ⊢ (𝑣 = 𝐴 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
15	14	exbidv 1922	. . 3 ⊢ (𝑣 = 𝐴 → (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
16	10, 15	mpbid 232	. 2 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
17	1, 16	vtocle 3510	1 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-cleq 2723  df-clel 2806  df-nfc 2881

This theorem is referenced by:  zfrep3cl  5230  zfrep4  5231