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Theorem sborv 1786
Description: Version of sbor 1844 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sborv ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sborv
StepHypRef Expression
1 sb5 1783 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓)))
2 andi 742 . . . 4 ((𝑥 = 𝑦 ∧ (𝜑𝜓)) ↔ ((𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜓)))
32exbii 1512 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓)) ↔ ∃𝑥((𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜓)))
4 19.43 1535 . . 3 (∃𝑥((𝑥 = 𝑦𝜑) ∨ (𝑥 = 𝑦𝜓)) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∨ ∃𝑥(𝑥 = 𝑦𝜓)))
51, 3, 43bitri 199 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∨ ∃𝑥(𝑥 = 𝑦𝜓)))
6 sb5 1783 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
7 sb5 1783 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑦𝜓))
86, 7orbi12i 691 . 2 (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∨ ∃𝑥(𝑥 = 𝑦𝜓)))
95, 8bitr4i 180 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wo 639  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbor  1844
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