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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.) |
Ref | Expression |
---|---|
sbequ1 | ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.4 582 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
2 | 19.8a 2039 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | df-sb 1868 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
4 | 1, 2, 3 | sylanbrc 695 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) |
5 | 4 | ex 449 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 |
This theorem is referenced by: sbequ12 2097 dfsb2 2361 sbequi 2363 sbi1 2380 2eu6 2546 sb5ALT 37752 2pm13.193 37789 2pm13.193VD 38161 sb5ALTVD 38171 |
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