Theorem axrep2 1474

Description: Axiom of Replacement slightly strengthened from axrep 1473; w may occur free in φ.

Assertion

Ref	Expression
axrep2	⊢ ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem axrep2

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 ⊢ (∃y∀z(φ → z = y) → ∀y∃y∀z(φ → z = y))
2		ax-17 925	. . . . . 6 ⊢ (z ∈ x → ∀y z ∈ x)
3		ax-17 925	. . . . . . . 8 ⊢ (x ∈ w → ∀y x ∈ w)
4		hba1 698	. . . . . . . 8 ⊢ (∀yφ → ∀y∀yφ)
5	3, 4	hban 704	. . . . . . 7 ⊢ ((x ∈ w ∧ ∀yφ) → ∀y(x ∈ w ∧ ∀yφ))
6	5	hbex 701	. . . . . 6 ⊢ (∃x(x ∈ w ∧ ∀yφ) → ∀y∃x(x ∈ w ∧ ∀yφ))
7	2, 6	hbbi 705	. . . . 5 ⊢ ((z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)) → ∀y(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))
8	7	hbal 700	. . . 4 ⊢ (∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)) → ∀y∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))
9	1, 8	hbim 702	. . 3 ⊢ ((∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))) → ∀y(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))))
10	9	hbex 701	. 2 ⊢ (∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))) → ∀y∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))))
11		visset 1350	. 2 ⊢ w ∈ V
12		a14b 820	. . . . . . . 8 ⊢ (y = w → (x ∈ y ↔ x ∈ w))
13	12	anbi1d 469	. . . . . . 7 ⊢ (y = w → ((x ∈ y ∧ ∀yφ) ↔ (x ∈ w ∧ ∀yφ)))
14	13	biexdv 936	. . . . . 6 ⊢ (y = w → (∃x(x ∈ y ∧ ∀yφ) ↔ ∃x(x ∈ w ∧ ∀yφ)))
15	14	bibi2d 470	. . . . 5 ⊢ (y = w → ((z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ)) ↔ (z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))))
16	15	bialdv 935	. . . 4 ⊢ (y = w → (∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ)) ↔ ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ))))
17	16	imbi2d 464	. . 3 ⊢ (y = w → ((∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ))) ↔ (∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))))
18	17	biexdv 936	. 2 ⊢ (y = w → (∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ))) ↔ ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))))
19		axrep 1473	. 2 ⊢ ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ)))
20	10, 11, 18, 19	vtoclf 1377	1 ⊢ ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ w ∧ ∀yφ)))

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

This theorem is referenced by:  zfrep2 1475

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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