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Mirrors > Home > MPE Home > Th. List > spsbbi | Structured version Visualization version GIF version |
Description: Biconditional property for substitution. Closed form of sbbii 2071. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2060. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
spsbbi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 214 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | alimi 1805 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
3 | spsbim 2067 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) |
5 | biimpr 219 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
6 | 5 | alimi 1805 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑)) |
7 | spsbim 2067 | . . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑)) |
9 | 4, 8 | impbid 211 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-sb 2060 |
This theorem is referenced by: sbbidv 2074 sbbid 2230 abbi 2794 sbeqi 37538 |
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