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Mirrors > Home > MPE Home > Th. List > Mathboxes > eleq2w2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of eleq2w2 2732 and special instance of eleq2 2819. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eleq2w2ALT | ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2729 | . . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
3 | eleq1w 2813 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
4 | eleq1w 2813 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | bibi12d 349 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
6 | 5 | spvv 2006 | . 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2728 df-clel 2809 |
This theorem is referenced by: (None) |
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