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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 6920 | . 2 ⊢ (𝐸‘𝑤) ∈ V | |
2 | fvex 6920 | . 2 ⊢ (𝐹‘𝑤) ∈ V | |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
4 | sbcie2s.a | . . . . . 6 ⊢ 𝐴 = (𝐸‘𝑊) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴) |
6 | 5 | eqeq2d 2746 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) ↔ 𝑎 = 𝐴)) |
7 | 6 | biimpd 229 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) → 𝑎 = 𝐴)) |
8 | fveq2 6907 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊)) | |
9 | sbcie2s.b | . . . . . 6 ⊢ 𝐵 = (𝐹‘𝑊) | |
10 | 8, 9 | eqtr4di 2793 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵) |
11 | 10 | eqeq2d 2746 | . . . 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 3876 | 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 1537 [wsbc 3791 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: isassa 21894 istrkgc 28477 istrkgb 28478 istrkge 28480 istrkgl 28481 ishpg 28782 iscgra 28832 |
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