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Theorem sbi2v 1865
 Description: Reverse direction of sbimv 1866. (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 1655 . . 3 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)) → ∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
2 pm3.3 259 . . . . 5 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦𝜓))))
3 pm2.04 82 . . . . 5 ((𝜑 → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
42, 3syli 37 . . . 4 (((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (𝑥 = 𝑦 → (𝜑𝜓)))
54alimi 1432 . . 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 1860 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 sb6 1859 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
97, 8imbi12i 238 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
10 sb6 1859 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
116, 9, 103imtr4i 200 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469  [wsb 1736 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-sb 1737 This theorem is referenced by:  sbimv  1866
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