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Theorem sbi1 2069
 Description: Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.)
Assertion
Ref Expression
sbi1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbi1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2063 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))))
2 ax-2 7 . . . . . 6 ((𝑥 = 𝑧 → (𝜑𝜓)) → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑧𝜓)))
32al2imi 1809 . . . . 5 (∀𝑥(𝑥 = 𝑧 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑧𝜓)))
43imim3i 64 . . . 4 ((𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))) → ((𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)) → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜓))))
54al2imi 1809 . . 3 (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))) → (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)) → ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜓))))
6 df-sb 2063 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
7 df-sb 2063 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜓)))
85, 6, 73imtr4g 297 . 2 (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
91, 8sylbi 218 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1528  [wsb 2062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803 This theorem depends on definitions:  df-bi 208  df-sb 2063 This theorem is referenced by:  spsbim  2070  sbimi  2072  sb2imi  2073  sbim  2305  2sb5ndVD  41109  2sb5ndALT  41131
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