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Mirrors > Home > ILE Home > Th. List > sbimi | 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 12 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)) |
3 | 1 | anim2i 340 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓)) |
4 | 3 | eximi 1593 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
5 | 2, 4 | anim12i 336 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
6 | df-sb 1756 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | df-sb 1756 | . 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
8 | 5, 6, 7 | 3imtr4i 200 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1485 [wsb 1755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-sb 1756 |
This theorem is referenced by: sbbii 1758 sb6f 1796 hbsb3 1801 sbidm 1844 sbco 1961 sbcocom 1963 sbalyz 1992 hbsb4t 2006 moimv 2085 elsb1 2148 elsb2 2149 oprcl 3787 peano1 4576 peano2 4577 |
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