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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcies | 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 | 
|---|---|
| sbcies.a | ⊢ 𝐴 = (𝐸‘𝑊) | 
| sbcies.1 | ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbcies | ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvexd 6920 | . 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V) | |
| 2 | simpr 484 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤)) | |
| 3 | sbcies.a | . . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊) | |
| 4 | fveq2 6905 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
| 5 | 3, 4 | eqtr4id 2795 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤)) | 
| 6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤)) | 
| 7 | 2, 6 | eqtr4d 2779 | . . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = 𝐴) | 
| 8 | sbcies.1 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜑 ↔ 𝜓)) | 
| 10 | 9 | bicomd 223 | . 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑)) | 
| 11 | 1, 10 | sbcied 3831 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 Vcvv 3479 [wsbc 3787 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: (None) | 
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