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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliminable2b | Structured version Visualization version GIF version |
Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eliminable2b | ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2818 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1534 = wceq 1536 ∈ wcel 2113 {cab 2802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2817 |
This theorem is referenced by: (None) |
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