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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssbid1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of bj-ssbid1 34501, not using sbequ1 2249. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-ssbid1ALT | ⊢ (𝜑 → [𝑥 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12v 2180 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | 1 | equcoms 2032 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | 2 | com12 32 | . . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | 3 | alrimiv 1934 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | df-sb 2075 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
6 | 4, 5 | sylibr 237 | 1 ⊢ (𝜑 → [𝑥 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 [wsb 2074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-sb 2075 |
This theorem is referenced by: (None) |
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