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Theorem sbi1v 1787
Description: Forward direction of sbimv 1789. (Contributed by Jim Kingdon, 25-Dec-2017.)
Assertion
Ref Expression
sbi1v ([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 sbi1v
StepHypRef Expression
1 sb6 1782 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 sb6 1782 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
3 ax-2 6 . . . . 5 ((𝑥 = 𝑦 → (𝜑𝜓)) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
43al2imi 1363 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
5 sb2 1666 . . . 4 (∀𝑥(𝑥 = 𝑦𝜓) → [𝑦 / 𝑥]𝜓)
64, 5syl6 33 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜓))
72, 6sylbi 118 . 2 ([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜓))
81, 7syl5bi 145 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  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:  sbimv  1789
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