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Mirrors > Home > MPE Home > Th. List > eleq2dALT | Structured version Visualization version GIF version |
Description: Alternate proof of eleq2d 2673, shorter at the expense of using more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eleq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eleq2dALT | ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1d.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | dfcleq 2604 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | sylib 207 | . . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | 3 | 19.21bi 2047 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
5 | 4 | anbi2d 736 | . . 3 ⊢ (𝜑 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
6 | 5 | exbidv 1837 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))) |
7 | df-clel 2606 | . 2 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴)) | |
8 | df-clel 2606 | . 2 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 302 | 1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: (None) |
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