Theorem sbexyz 1876

Description: Move existential quantifier in and out of substitution. Identical to sbex 1877 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)

Assertion

Ref	Expression
sbexyz	⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)

Distinct variable group:   x,y,z

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

Proof of Theorem sbexyz

Step	Hyp	Ref	Expression
1		sb5 1764	. . 3 ⊢ ([z / y]∃xφ ↔ ∃y(y = z ∧ ∃xφ))
2		exdistr 1784	. . 3 ⊢ (∃y∃x(y = z ∧ φ) ↔ ∃y(y = z ∧ ∃xφ))
3		excom 1551	. . 3 ⊢ (∃y∃x(y = z ∧ φ) ↔ ∃x∃y(y = z ∧ φ))
4	1, 2, 3	3bitr2i 197	. 2 ⊢ ([z / y]∃xφ ↔ ∃x∃y(y = z ∧ φ))
5		sb5 1764	. . 3 ⊢ ([z / y]φ ↔ ∃y(y = z ∧ φ))
6	5	exbii 1493	. 2 ⊢ (∃x[z / y]φ ↔ ∃x∃y(y = z ∧ φ))
7	4, 6	bitr4i 176	1 ⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sbex  1877