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Mirrors > Home > MPE Home > Th. List > sbequ1 | Structured version Visualization version GIF version |
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
sbequ1 | ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equeucl 2031 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑦 = 𝑡 → 𝑥 = 𝑦)) | |
2 | ax12v 2176 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 1, 2 | syl6 35 | . . . 4 ⊢ (𝑥 = 𝑡 → (𝑦 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
4 | 3 | com23 86 | . . 3 ⊢ (𝑥 = 𝑡 → (𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
5 | 4 | alrimdv 1930 | . 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
6 | df-sb 2070 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
7 | 5, 6 | syl6ibr 255 | 1 ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 |
This theorem is referenced by: sbequ12 2250 dfsb1 2499 dfsb2 2511 sbi1OLD 2519 2eu6 2719 bj-ssbid1 34110 sb5ALT 41231 2pm13.193 41258 2pm13.193VD 41609 sb5ALTVD 41619 |
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