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Theorem sbanv 1785
Description: Version of sban 1845 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 1782 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 sb6 1782 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
3 sb6 1782 . . . 4 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
42, 3anbi12i 441 . . 3 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜓)))
5 19.26 1386 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜓)))
6 pm4.76 546 . . . 4 (((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ (𝑥 = 𝑦 → (𝜑𝜓)))
76albii 1375 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) ∧ (𝑥 = 𝑦𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
84, 5, 73bitr2i 201 . 2 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
91, 8bitr4i 180 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  [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-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:  sban  1845
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