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Mirrors > Home > MPE Home > Th. List > snjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-sn 4591. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
snjust | ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2737 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) | |
2 | 1 | cbvabv 2806 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑧 ∣ 𝑧 = 𝐴} |
3 | eqeq1 2737 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴)) | |
4 | 3 | cbvabv 2806 | . 2 ⊢ {𝑧 ∣ 𝑧 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
5 | 2, 4 | eqtri 2761 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 |
This theorem is referenced by: (None) |
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