This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2sb5

Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005)

Ref Expression
Assertion 2sb5 ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )

Proof

Step Hyp Ref Expression
1 sb5 ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ [ 𝑤 / 𝑦 ] 𝜑 ) )
2 19.42v ⊢ ( ∃ 𝑦 ( 𝑥 = 𝑧 ∧ ( 𝑦 = 𝑤 ∧ 𝜑 ) ) ↔ ( 𝑥 = 𝑧 ∧ ∃ 𝑦 ( 𝑦 = 𝑤 ∧ 𝜑 ) ) )
3 anass ⊢ ( ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) ↔ ( 𝑥 = 𝑧 ∧ ( 𝑦 = 𝑤 ∧ 𝜑 ) ) )
4 3 exbii ⊢ ( ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑥 = 𝑧 ∧ ( 𝑦 = 𝑤 ∧ 𝜑 ) ) )
5 sb5 ⊢ ( [ 𝑤 / 𝑦 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑤 ∧ 𝜑 ) )
6 5 anbi2i ⊢ ( ( 𝑥 = 𝑧 ∧ [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝑥 = 𝑧 ∧ ∃ 𝑦 ( 𝑦 = 𝑤 ∧ 𝜑 ) ) )
7 2 4 6 3bitr4ri ⊢ ( ( 𝑥 = 𝑧 ∧ [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )
8 7 exbii ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑧 ∧ [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )
9 1 8 bitri ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )