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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 6113 | . 2 ⊢ (𝐸‘𝑤) ∈ V | |
2 | fvex 6113 | . 2 ⊢ (𝐹‘𝑤) ∈ V | |
3 | simprl 790 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = (𝐸‘𝑤)) | |
4 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
5 | sbcie2s.a | . . . . . . . 8 ⊢ 𝐴 = (𝐸‘𝑊) | |
6 | 4, 5 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴) |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐸‘𝑤) = 𝐴) |
8 | 3, 7 | eqtrd 2644 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = 𝐴) |
9 | simprr 792 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = (𝐹‘𝑤)) | |
10 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊)) | |
11 | sbcie2s.b | . . . . . . . 8 ⊢ 𝐵 = (𝐹‘𝑊) | |
12 | 10, 11 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐹‘𝑤) = 𝐵) |
14 | 9, 13 | eqtrd 2644 | . . . . 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 691 | . . . 4 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜑 ↔ 𝜓)) |
17 | 16 | bicomd 212 | . . 3 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜓 ↔ 𝜑)) |
18 | 17 | ex 449 | . 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜓 ↔ 𝜑))) |
19 | 1, 2, 18 | sbc2iedv 3473 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 [wsbc 3402 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: istrkgc 25153 istrkgb 25154 istrkge 25156 istrkgl 25157 ishpg 25451 iscgra 25501 |
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