Metamath Proof Explorer


Theorem axrep4

Description: A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994) (Proof shortened by Matthew House, 18-Sep-2025)

Ref Expression
Hypothesis axrep4.1 𝑧 𝜑
Assertion axrep4 ( ∀ 𝑥𝑧𝑦 ( 𝜑𝑦 = 𝑧 ) → ∃ 𝑧𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 axrep4.1 𝑧 𝜑
2 ax-rep ( ∀ 𝑥𝑧𝑦 ( ∀ 𝑧 𝜑𝑦 = 𝑧 ) → ∃ 𝑧𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ) )
3 1 19.3 ( ∀ 𝑧 𝜑𝜑 )
4 3 imbi1i ( ( ∀ 𝑧 𝜑𝑦 = 𝑧 ) ↔ ( 𝜑𝑦 = 𝑧 ) )
5 4 albii ( ∀ 𝑦 ( ∀ 𝑧 𝜑𝑦 = 𝑧 ) ↔ ∀ 𝑦 ( 𝜑𝑦 = 𝑧 ) )
6 5 exbii ( ∃ 𝑧𝑦 ( ∀ 𝑧 𝜑𝑦 = 𝑧 ) ↔ ∃ 𝑧𝑦 ( 𝜑𝑦 = 𝑧 ) )
7 6 albii ( ∀ 𝑥𝑧𝑦 ( ∀ 𝑧 𝜑𝑦 = 𝑧 ) ↔ ∀ 𝑥𝑧𝑦 ( 𝜑𝑦 = 𝑧 ) )
8 3 anbi2i ( ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ↔ ( 𝑥𝑤𝜑 ) )
9 8 exbii ( ∃ 𝑥 ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) )
10 9 bibi2i ( ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) ) )
11 10 albii ( ∀ 𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) ) )
12 11 exbii ( ∃ 𝑧𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤 ∧ ∀ 𝑧 𝜑 ) ) ↔ ∃ 𝑧𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) ) )
13 2 7 12 3imtr3i ( ∀ 𝑥𝑧𝑦 ( 𝜑𝑦 = 𝑧 ) → ∃ 𝑧𝑦 ( 𝑦𝑧 ↔ ∃ 𝑥 ( 𝑥𝑤𝜑 ) ) )