| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sbcie2s | Structured version Visualization version GIF version | ||
| Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.) |
| Ref | Expression |
|---|---|
| sbcie2s.a | ⊢ 𝐴 = (𝐸‘𝑊) |
| sbcie2s.b | ⊢ 𝐵 = (𝐹‘𝑊) |
| sbcie2s.1 | ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| sbcie2s | ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvex 6919 | . 2 ⊢ (𝐸‘𝑤) ∈ V | |
| 2 | fvex 6919 | . 2 ⊢ (𝐹‘𝑤) ∈ V | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
| 4 | sbcie2s.a | . . . . . 6 ⊢ 𝐴 = (𝐸‘𝑊) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴) |
| 6 | 5 | eqeq2d 2748 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) ↔ 𝑎 = 𝐴)) |
| 7 | 6 | biimpd 229 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) → 𝑎 = 𝐴)) |
| 8 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊)) | |
| 9 | sbcie2s.b | . . . . . 6 ⊢ 𝐵 = (𝐹‘𝑊) | |
| 10 | 8, 9 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵) |
| 11 | 10 | eqeq2d 2748 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑏 = (𝐹‘𝑤) ↔ 𝑏 = 𝐵)) |
| 12 | 11 | biimpd 229 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑏 = (𝐹‘𝑤) → 𝑏 = 𝐵)) |
| 13 | sbcie2s.1 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 14 | 13 | a1i 11 | . . 3 ⊢ (𝑤 = 𝑊 → ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))) |
| 15 | 7, 12, 14 | syl2and 608 | . 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜑 ↔ 𝜓))) |
| 16 | 1, 2, 15 | sbc2iedv 3867 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 [wsbc 3788 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: isassa 21876 istrkgc 28462 istrkgb 28463 istrkge 28465 istrkgl 28466 ishpg 28767 iscgra 28817 |
| Copyright terms: Public domain | W3C validator |