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Mirrors > Home > MPE Home > Th. List > sbimiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbimi 2078 as of 6-Jul-2023. Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbimiOLD.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
sbimiOLD | ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbimiOLD.1 | . . . 4 ⊢ (𝜑 → 𝜓) | |
2 | 1 | imim2i 16 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)) |
3 | 1 | anim2i 618 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓)) |
4 | 3 | eximi 1834 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
5 | 2, 4 | anim12i 614 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
6 | dfsb1 2509 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | dfsb1 2509 | . 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
8 | 5, 6, 7 | 3imtr4i 294 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1779 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: (None) |
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