Theorem zfrepclf 5241

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 2941	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = 𝐴
4		eleq2 2851	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ 𝐴))
5		zfrepclf.3	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))
6	4, 5	biimtrdi 255	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
7	3, 6	alrimi 2248	. . . 4 ⊢ (𝑣 = 𝐴 → ∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
8		nfv 1934	. . . . 5 ⊢ Ⅎ𝑧𝜑
9	8	axrep5 5235	. . . 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 640	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑥 ∈ 𝑣 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
12	3, 11	exbid 2258	. . . . . 6 ⊢ (𝑣 = 𝐴 → (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
13	12	bibi2d 344	. . . . 5 ⊢ (𝑣 = 𝐴 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
14	13	albidv 1940	. . . 4 ⊢ (𝑣 = 𝐴 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
15	14	exbidv 1941	. . 3 ⊢ (𝑣 = 𝐴 → (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))))
16	10, 15	mpbid 234	. 2 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
17	1, 16	vtocle 3523	1 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142  Ⅎwnfc 2909  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-cleq 2754  df-clel 2837  df-nfc 2911

This theorem is referenced by:  zfrep3cl  5242  zfrep4  5243