Theorem sbexyz 1879

Description: Move existential quantifier in and out of substitution. Identical to sbex 1880 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 1767	. . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
2		exdistr 1787	. . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
3		excom 1554	. . 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 1767	. . 3 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))
6	5	exbii 1496	. 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 1381  [wsb 1645

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbex  1880