Theorem zfrep3 1476

Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us φ is analogous to a "function" from x to y (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set z that corresponds to the "image" of φ restricted to some other set w. The hypothesis says z must not be free in φ.

Hypothesis

Ref	Expression
zfrep3.1	⊢ (φ → ∀zφ)

Assertion

Ref	Expression
zfrep3	⊢ (∀x(x ∈ w → ∃z∀y(φ → y = z)) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem zfrep3

Step	Hyp	Ref	Expression
1		19.37v 961	. . . . 5 ⊢ (∃z(x ∈ w → ∀y(φ → y = z)) ↔ (x ∈ w → ∃z∀y(φ → y = z)))
2		impexp 276	. . . . . . . 8 ⊢ (((x ∈ w ∧ φ) → y = z) ↔ (x ∈ w → (φ → y = z)))
3	2	bial 695	. . . . . . 7 ⊢ (∀y((x ∈ w ∧ φ) → y = z) ↔ ∀y(x ∈ w → (φ → y = z)))
4		19.21v 942	. . . . . . 7 ⊢ (∀y(x ∈ w → (φ → y = z)) ↔ (x ∈ w → ∀y(φ → y = z)))
5	3, 4	bitr2 152	. . . . . 6 ⊢ ((x ∈ w → ∀y(φ → y = z)) ↔ ∀y((x ∈ w ∧ φ) → y = z))
6	5	biex 733	. . . . 5 ⊢ (∃z(x ∈ w → ∀y(φ → y = z)) ↔ ∃z∀y((x ∈ w ∧ φ) → y = z))
7	1, 6	bitr3 153	. . . 4 ⊢ ((x ∈ w → ∃z∀y(φ → y = z)) ↔ ∃z∀y((x ∈ w ∧ φ) → y = z))
8	7	bial 695	. . 3 ⊢ (∀x(x ∈ w → ∃z∀y(φ → y = z)) ↔ ∀x∃z∀y((x ∈ w ∧ φ) → y = z))
9		ax-17 925	. . . . 5 ⊢ (x ∈ w → ∀z x ∈ w)
10		zfrep3.1	. . . . 5 ⊢ (φ → ∀zφ)
11	9, 10	hban 704	. . . 4 ⊢ ((x ∈ w ∧ φ) → ∀z(x ∈ w ∧ φ))
12	11	zfrep2 1475	. . 3 ⊢ (∀x∃z∀y((x ∈ w ∧ φ) → y = z) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ (x ∈ w ∧ φ))))
13	8, 12	sylbi 174	. 2 ⊢ (∀x(x ∈ w → ∃z∀y(φ → y = z)) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ (x ∈ w ∧ φ))))
14		anabs5 375	. . . . . 6 ⊢ ((x ∈ w ∧ (x ∈ w ∧ φ)) ↔ (x ∈ w ∧ φ))
15	14	biex 733	. . . . 5 ⊢ (∃x(x ∈ w ∧ (x ∈ w ∧ φ)) ↔ ∃x(x ∈ w ∧ φ))
16	15	bibi2i 460	. . . 4 ⊢ ((y ∈ z ↔ ∃x(x ∈ w ∧ (x ∈ w ∧ φ))) ↔ (y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
17	16	bial 695	. . 3 ⊢ (∀y(y ∈ z ↔ ∃x(x ∈ w ∧ (x ∈ w ∧ φ))) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
18	17	biex 733	. 2 ⊢ (∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ (x ∈ w ∧ φ))) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ w ∧ φ)))
19	13, 18	sylib 173	1 ⊢ (∀x(x ∈ w → ∃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:  zfrepclf 1477

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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