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Mirrors > Home > MPE Home > Th. List > spsbim | Structured version Visualization version GIF version |
Description: Distribute substitution over implication. Closed form of sbimi 2082. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2073. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.) |
Ref | Expression |
---|---|
spsbim | ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2076 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓)) | |
2 | sbi1 2079 | . 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 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 ax-5 1918 |
This theorem depends on definitions: df-bi 210 df-sb 2073 |
This theorem is referenced by: spsbbi 2081 sbimdv 2086 sbimd 2244 mo3 2563 bj-hbsb3t 34656 wl-mo3t 35417 pm11.59 41623 sbiota1 41666 |
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