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Mirrors > Home > MPE Home > Th. List > sbimi | Structured version Visualization version GIF version |
Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Ref | Expression |
---|---|
sbimi.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
sbimi | ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbimi.1 | . . . 4 ⊢ (𝜑 → 𝜓) | |
2 | 1 | imim2i 16 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)) |
3 | 1 | anim2i 610 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓)) |
4 | 3 | eximi 1878 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
5 | 2, 4 | anim12i 606 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
6 | df-sb 2012 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | df-sb 2012 | . 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
8 | 5, 6, 7 | 3imtr4i 284 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∃wex 1823 [wsb 2011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-sb 2012 |
This theorem is referenced by: sbbii 2019 sbi2v 2279 sbiev 2286 hbsb3 2440 sb6f 2461 sbi2 2469 sbie 2484 2mo 2678 fmptdF 30038 funcnv4mpt 30051 disjdsct 30063 measiuns 30886 ballotlemodife 31166 bj-hbsb3v 33351 bj-sbidmOLD 33414 mptsnunlem 33788 |
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