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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssbequ | Structured version Visualization version GIF version |
Description: Equality property for substitution, from Tarski's system. Compare sbequ 2376. (Contributed by BJ, 30-Dec-2020.) |
Ref | Expression |
---|---|
bj-ssbequ | ⊢ (𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 1953 | . . . 4 ⊢ (𝑠 = 𝑡 → (𝑦 = 𝑠 ↔ 𝑦 = 𝑡)) | |
2 | 1 | imbi1d 331 | . . 3 ⊢ (𝑠 = 𝑡 → ((𝑦 = 𝑠 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
3 | 2 | albidv 1849 | . 2 ⊢ (𝑠 = 𝑡 → (∀𝑦(𝑦 = 𝑠 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
4 | df-ssb 32620 | . 2 ⊢ ([𝑠/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑠 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | df-ssb 32620 | . 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
6 | 3, 4, 5 | 3bitr4g 303 | 1 ⊢ (𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 [wssb 32619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-ssb 32620 |
This theorem is referenced by: (None) |
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