Theorem sbexyz 2015

Description: Move existential quantifier in and out of substitution. Identical to sbex 2016 except that it has an additional disjoint variable condition on 𝑦, 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 1899	. . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
2		exdistr 1921	. . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
3		excom 1675	. . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑))
4	1, 2, 3	3bitr2i 208	. 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑))
5		sb5 1899	. . 3 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))
6	5	exbii 1616	. 2 ⊢ (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑))
7	4, 6	bitr4i 187	1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1503  [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  sbex  2016