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Theorem 2pm13.193 42172
Description: pm13.193 42029 for two variables. pm13.193 42029 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 42523. (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 764 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑥 = 𝑢)
2 simplr 766 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑦 = 𝑣)
3 simpr 485 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4 sbequ2 2241 . . . . 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 2241 . . . 4 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
72, 5, 6sylc 65 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝜑)
81, 2, 7jca31 515 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
9 simpll 764 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝑥 = 𝑢)
10 simplr 766 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝑦 = 𝑣)
11 simpr 485 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
12 sbequ1 2240 . . . . 5 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
1310, 11, 12sylc 65 . . . 4 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → [𝑣 / 𝑦]𝜑)
14 sbequ1 2240 . . . 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 515 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
178, 16impbii 208 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068
This theorem is referenced by:  2sb5nd  42180  2sb5ndVD  42530  2sb5ndALT  42552
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