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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.) |
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]𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6683 | . 2 ⊢ (𝐸‘𝑤) ∈ V | |
2 | fvex 6683 | . 2 ⊢ (𝐹‘𝑤) ∈ V | |
3 | simprl 769 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = (𝐸‘𝑤)) | |
4 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
5 | sbcie2s.a | . . . . . . . 8 ⊢ 𝐴 = (𝐸‘𝑊) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴) |
7 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐸‘𝑤) = 𝐴) |
8 | 3, 7 | eqtrd 2856 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = 𝐴) |
9 | simprr 771 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = (𝐹‘𝑤)) | |
10 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊)) | |
11 | sbcie2s.b | . . . . . . . 8 ⊢ 𝐵 = (𝐹‘𝑊) | |
12 | 10, 11 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵) |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐹‘𝑤) = 𝐵) |
14 | 9, 13 | eqtrd 2856 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = 𝐵) |
15 | sbcie2s.1 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)) | |
16 | 8, 14, 15 | syl2anc 586 | . . . 4 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜑 ↔ 𝜓)) |
17 | 16 | bicomd 225 | . . 3 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜓 ↔ 𝜑)) |
18 | 17 | ex 415 | . 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜓 ↔ 𝜑))) |
19 | 1, 2, 18 | sbc2iedv 3851 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 [wsbc 3772 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 |
This theorem is referenced by: istrkgc 26240 istrkgb 26241 istrkge 26243 istrkgl 26244 ishpg 26545 iscgra 26595 |
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