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Mirrors > Home > MPE Home > Th. List > nfceqdfOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfceqdf 2892 as of 23-Aug-2024. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfceqdf.1 | ⊢ Ⅎ𝑥𝜑 |
nfceqdf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nfceqdfOLD | ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqdf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | nfceqdf.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2813 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | nfbidf 2209 | . . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)) |
5 | 4 | albidv 1915 | . 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)) |
6 | df-nfc 2879 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
7 | df-nfc 2879 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-cleq 2718 df-clel 2804 df-nfc 2879 |
This theorem is referenced by: (None) |
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