Theorem zfrep2 1475

Description: A more traditional version of the Axiom of Replacement.

Hypothesis

Ref	Expression
zfrep2.1	⊢ (φ → ∀zφ)

Assertion

Ref	Expression
zfrep2	⊢ (∀x∃z∀y(φ → y = z) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem zfrep2

Step	Hyp	Ref	Expression
1		axrep2 1474	. . 3 ⊢ ∃x(∃z∀y(φ → y = z) → ∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)))
2	1	19.35i 755	. 2 ⊢ (∀x∃z∀y(φ → y = z) → ∃x∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)))
3		ax-17 925	. . . . 5 ⊢ (y ∈ x → ∀z y ∈ x)
4		ax-17 925	. . . . . . 7 ⊢ (x ∈ w → ∀z x ∈ w)
5		hba1 698	. . . . . . 7 ⊢ (∀zφ → ∀z∀zφ)
6	4, 5	hban 704	. . . . . 6 ⊢ ((x ∈ w ∧ ∀zφ) → ∀z(x ∈ w ∧ ∀zφ))
7	6	hbex 701	. . . . 5 ⊢ (∃x(x ∈ w ∧ ∀zφ) → ∀z∃x(x ∈ w ∧ ∀zφ))
8	3, 7	hbbi 705	. . . 4 ⊢ ((y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)) → ∀z(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)))
9	8	hbal 700	. . 3 ⊢ (∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)) → ∀z∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)))
10		ax-17 925	. . . . 5 ⊢ (y ∈ z → ∀x y ∈ z)
11		hbe1 709	. . . . 5 ⊢ (∃x(x ∈ w ∧ φ) → ∀x∃x(x ∈ w ∧ φ))
12	10, 11	hbbi 705	. . . 4 ⊢ ((y ∈ z ↔ ∃x(x ∈ w ∧ φ)) → ∀x(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
13	12	hbal 700	. . 3 ⊢ (∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)) → ∀x∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
14		ax-17 925	. . . 4 ⊢ (x = z → ∀y x = z)
15		a14b 820	. . . . 5 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
16		ax-4 673	. . . . . . . . 9 ⊢ (∀zφ → φ)
17		zfrep2.1	. . . . . . . . 9 ⊢ (φ → ∀zφ)
18	16, 17	impbi 139	. . . . . . . 8 ⊢ (∀zφ ↔ φ)
19	18	anbi2i 367	. . . . . . 7 ⊢ ((x ∈ w ∧ ∀zφ) ↔ (x ∈ w ∧ φ))
20	19	biex 733	. . . . . 6 ⊢ (∃x(x ∈ w ∧ ∀zφ) ↔ ∃x(x ∈ w ∧ φ))
21	20	a1i 7	. . . . 5 ⊢ (x = z → (∃x(x ∈ w ∧ ∀zφ) ↔ ∃x(x ∈ w ∧ φ)))
22	15, 21	bibi12d 477	. . . 4 ⊢ (x = z → ((y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)) ↔ (y ∈ z ↔ ∃x(x ∈ w ∧ φ))))
23	14, 22	biald 782	. . 3 ⊢ (x = z → (∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ))))
24	9, 13, 23	cbvex 849	. 2 ⊢ (∃x∀y(y ∈ x ↔ ∃x(x ∈ w ∧ ∀zφ)) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
25	2, 24	sylib 173	1 ⊢ (∀x∃z∀y(φ → y = z) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803

This theorem is referenced by:  zfrep3 1476  funimaexg 2715

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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