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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 592 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ 𝜓)) |
4 | 3 | eximi 1802 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
5 | 2, 4 | anim12i 589 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
6 | df-sb 1938 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | df-sb 1938 | . 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
8 | 5, 6, 7 | 3imtr4i 281 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1744 [wsb 1937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 df-sb 1938 |
This theorem is referenced by: sbbii 1944 hbsb3 2392 sb6f 2413 sbi2 2421 sbie 2436 2mo 2580 fmptdF 29584 funcnv4mpt 29598 disjdsct 29608 measiuns 30408 ballotlemodife 30687 bj-hbsb3v 32886 bj-sbidmOLD 32956 mptsnunlem 33315 |
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