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Theorem sbi2v 1814
Description: Reverse direction of sbimv 1815. (Contributed by Jim Kingdon, 18-Jan-2018.)
Assertion
Ref Expression
sbi2v (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbi2v
StepHypRef Expression
1 19.38 1607 . . 3 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)) → ∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
2 pm3.3 257 . . . . 5 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦𝜓))))
3 pm2.04 81 . . . . 5 ((𝜑 → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
42, 3syli 37 . . . 4 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
54alimi 1385 . . 3 (∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
61, 5syl 14 . 2 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
7 sb5 1809 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 sb6 1808 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
97, 8imbi12i 237 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
10 sb6 1808 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
116, 9, 103imtr4i 199 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283  wex 1422  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbimv  1815
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