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Theorem 2pm13.193 38594
Description: pm13.193 38438 for two variables. pm13.193 38438 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 38965. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2pm13.193 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Proof of Theorem 2pm13.193
StepHypRef Expression
1 simpll 790 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑥 = 𝑢)
2 simplr 792 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑦 = 𝑣)
3 simpr 477 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4 sbequ2 1881 . . . . 5 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
51, 3, 4sylc 65 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑣 / 𝑦]𝜑)
6 sbequ2 1881 . . . 4 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
72, 5, 6sylc 65 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝜑)
81, 2, 7jca31 557 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
9 simpll 790 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝑥 = 𝑢)
10 simplr 792 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝑦 = 𝑣)
11 simpr 477 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
12 sbequ1 2109 . . . . 5 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
1310, 11, 12sylc 65 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → [𝑣 / 𝑦]𝜑)
14 sbequ1 2109 . . . 4 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
159, 13, 14sylc 65 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
169, 10, 15jca31 557 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
178, 16impbii 199 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  [wsb 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-sb 1880
This theorem is referenced by:  2sb5nd  38602  2sb5ndVD  38972  2sb5ndALT  38994
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