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Theorem sbanv 1843
Description: Version of sban 1904 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
Assertion
Ref Expression
sbanv ([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 sbanv
StepHypRef Expression
1 sb6 1840 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 sb6 1840 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
3 sb6 1840 . . . 4 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
42, 3anbi12i 453 . . 3 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜓)))
5 19.26 1440 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜓)))
6 pm4.76 576 . . . 4 (((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ (𝑥 = 𝑦 → (𝜑𝜓)))
76albii 1429 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
84, 5, 73bitr2i 207 . 2 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
91, 8bitr4i 186 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-sb 1719
This theorem is referenced by:  sban  1904
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