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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cleq | Structured version Visualization version GIF version |
Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.) |
Ref | Expression |
---|---|
bj-cleq | ⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1 5891 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | |
2 | eleq2 2878 | . . 3 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶))) | |
3 | 2 | alrimiv 1928 | . 2 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶))) |
4 | abbi1 2861 | . 2 ⊢ (∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)) → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)}) | |
5 | 1, 3, 4 | 3syl 18 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 {csn 4525 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: (None) |
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