Theorem axrep5 5214

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. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
axrep5.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
axrep5	⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep5

Step	Hyp	Ref	Expression
1		19.37v 2004	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑤 → ∀𝑦(𝜑 → 𝑦 = 𝑧)) ↔ (𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)))
2		impexp 451	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧) ↔ (𝑥 ∈ 𝑤 → (𝜑 → 𝑦 = 𝑧)))
3	2	albii 1826	. . . . . . 7 ⊢ (∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧) ↔ ∀𝑦(𝑥 ∈ 𝑤 → (𝜑 → 𝑦 = 𝑧)))
4		19.21v 1946	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝑤 → (𝜑 → 𝑦 = 𝑧)) ↔ (𝑥 ∈ 𝑤 → ∀𝑦(𝜑 → 𝑦 = 𝑧)))
5	3, 4	bitr2i 277	. . . . . 6 ⊢ ((𝑥 ∈ 𝑤 → ∀𝑦(𝜑 → 𝑦 = 𝑧)) ↔ ∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧))
6	5	exbii 1855	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑤 → ∀𝑦(𝜑 → 𝑦 = 𝑧)) ↔ ∃𝑧∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧))
7	1, 6	bitr3i 278	. . . 4 ⊢ ((𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) ↔ ∃𝑧∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧))
8	7	albii 1826	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) ↔ ∀𝑥∃𝑧∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧))
9		nfv 1921	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤
10		axrep5.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
11	9, 10	nfan 1906	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ 𝜑)
12	11	axrep4 5212	. . 3 ⊢ (∀𝑥∃𝑧∀𝑦((𝑥 ∈ 𝑤 ∧ 𝜑) → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑))))
13	8, 12	sylbi 218	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑))))
14		anabs5 669	. . . . . 6 ⊢ ((𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
15	14	exbii 1855	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
16	15	bibi2i 338	. . . 4 ⊢ ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑))) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
17	16	albii 1826	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑))) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
18	17	exbii 1855	. 2 ⊢ (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑤 ∧ 𝜑))) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
19	13, 18	sylib 219	1 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791

This theorem is referenced by:  zfrepclf  5220