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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 6203 | . 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V) | |
2 | simpr 477 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤)) | |
3 | fveq2 6191 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
4 | sbcies.a | . . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊) | |
5 | 3, 4 | syl6reqr 2675 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤)) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤)) |
7 | 2, 6 | eqtr4d 2659 | . . . 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 213 | . 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑)) |
11 | 1, 10 | sbcied 3472 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Vcvv 3200 [wsbc 3435 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 |
This theorem is referenced by: (None) |
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