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Mirrors > Home > MPE Home > Th. List > cbvraldvaOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvraldva 3231 as of 9-Mar-2025. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvrexdva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvraldvaOLD | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvrexdva.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | eqidd 2728 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐴) | |
3 | 1, 2 | cbvraldva2 3339 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wral 3056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-cleq 2719 df-clel 2805 df-ral 3057 |
This theorem is referenced by: (None) |
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