Theorem zfrepclf 2704

Description: An inference rule based on the Axiom of Replacement. Typically, φ defines a function from x to y.

Hypotheses

Ref	Expression
zfrepclf.1	⊢ (w ∈ A → ∀x w ∈ A)
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   w,A   x,y,z   x,w

Proof of Theorem zfrepclf

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 ⊢ A ∈ V
2		ax-17 973	. . . . . 6 ⊢ (w ∈ v → ∀x w ∈ v)
3		zfrepclf.1	. . . . . 6 ⊢ (w ∈ A → ∀x w ∈ A)
4	2, 3	hbeq 1568	. . . . 5 ⊢ (v = A → ∀x v = A)
5		eleq2 1538	. . . . . 6 ⊢ (v = A → (x ∈ v ↔ x ∈ A))
6		zfrepclf.3	. . . . . 6 ⊢ (x ∈ A → ∃z∀y(φ → y = z))
7	5, 6	syl6bi 214	. . . . 5 ⊢ (v = A → (x ∈ v → ∃z∀y(φ → y = z)))
8	4, 7	19.21ai 1000	. . . 4 ⊢ (v = A → ∀x(x ∈ v → ∃z∀y(φ → y = z)))
9		ax-17 973	. . . . 5 ⊢ (φ → ∀zφ)
10	9	axrep5 2703	. . . 4 ⊢ (∀x(x ∈ v → ∃z∀y(φ → y = z)) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ v ⋀ φ)))
11	8, 10	syl 10	. . 3 ⊢ (v = A → ∃z∀y(y ∈ z ↔ ∃x(x ∈ v ⋀ φ)))
12	5	anbi1d 619	. . . . . . 7 ⊢ (v = A → ((x ∈ v ⋀ φ) ↔ (x ∈ A ⋀ φ)))
13	4, 12	exbid 1107	. . . . . 6 ⊢ (v = A → (∃x(x ∈ v ⋀ φ) ↔ ∃x(x ∈ A ⋀ φ)))
14	13	bibi2d 620	. . . . 5 ⊢ (v = A → ((y ∈ z ↔ ∃x(x ∈ v ⋀ φ)) ↔ (y ∈ z ↔ ∃x(x ∈ A ⋀ φ))))
15	14	albidv 1280	. . . 4 ⊢ (v = A → (∀y(y ∈ z ↔ ∃x(x ∈ v ⋀ φ)) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ A ⋀ φ))))
16	15	exbidv 1281	. . 3 ⊢ (v = A → (∃z∀y(y ∈ z ↔ ∃x(x ∈ v ⋀ φ)) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ⋀ φ))))
17	11, 16	mpbid 195	. 2 ⊢ (v = A → ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ⋀ φ)))
18	1, 17	vtocle 1861	1 ⊢ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ⋀ φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814

This theorem is referenced by:  zfrep3cl 2705  zfrep4 2706

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462  ax-rep 2698

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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