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Mirrors > Home > MPE Home > Th. List > sbi1 | Structured version Visualization version GIF version |
Description: Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.) |
Ref | Expression |
---|---|
sbi1 | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2073 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑 → 𝜓)))) | |
2 | ax-2 7 | . . . . . 6 ⊢ ((𝑥 = 𝑧 → (𝜑 → 𝜓)) → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑧 → 𝜓))) | |
3 | 2 | al2imi 1823 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝑧 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑧 → 𝜓))) |
4 | 3 | imim3i 64 | . . . 4 ⊢ ((𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑 → 𝜓))) → ((𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜓)))) |
5 | 4 | al2imi 1823 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑 → 𝜓))) → (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) → ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜓)))) |
6 | df-sb 2073 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
7 | df-sb 2073 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜓))) | |
8 | 5, 6, 7 | 3imtr4g 299 | . 2 ⊢ (∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → (𝜑 → 𝜓))) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
9 | 1, 8 | sylbi 220 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([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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