Theorem sbel2x 2459

Description: Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)

Assertion

Ref	Expression
sbel2x	⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Distinct variable group:   𝑥,𝑦,𝜑

Allowed substitution hints:   𝜑(𝑧,𝑤)

Proof of Theorem sbel2x

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfv 1843	. . 3 ⊢ Ⅎ𝑥𝜑
3	1, 2	2sb5rf 2451	. 2 ⊢ (𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
4		ancom 466	. . . 4 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
5	4	anbi1i 731	. . 3 ⊢ (((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
6	5	2exbii 1775	. 2 ⊢ (∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑦∃𝑥((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
7		excom 2042	. 2 ⊢ (∃𝑦∃𝑥((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
8	3, 6, 7	3bitri 286	1 ⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)