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Theorem sbi1 2079
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 2073 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))))
2 ax-2 7 . . . . . 6 ((𝑥 = 𝑧 → (𝜑𝜓)) → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑧𝜓)))
32al2imi 1823 . . . . 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 1823 . . 3 (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))) → (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)) → ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜓))))
6 df-sb 2073 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜑)))
7 df-sb 2073 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧𝜓)))
85, 6, 73imtr4g 299 . 2 (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑𝜓))) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
91, 8sylbi 220 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1541  [wsb 2072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-sb 2073
This theorem is referenced by:  spsbim  2080  sbimi  2082  sb2imi  2083  sbim  2306  sbcim1  3767  2sb5ndVD  42251  2sb5ndALT  42273
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