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Mirrors > Home > MPE Home > Th. List > Mathboxes > fveleq | Structured version Visualization version GIF version |
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
Ref | Expression |
---|---|
fveleq | ⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6767 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) | |
2 | 1 | eleq1d 2823 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) ∈ 𝑃 ↔ (𝐹‘𝐵) ∈ 𝑃)) |
3 | 2 | imbi2d 341 | 1 ⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-iota 6385 df-fv 6435 |
This theorem is referenced by: findfvcl 34627 |
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