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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 6684 | . 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V) | |
2 | simpr 487 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤)) | |
3 | fveq2 6669 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊)) | |
4 | sbcies.a | . . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊) | |
5 | 3, 4 | syl6reqr 2875 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤)) |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤)) |
7 | 2, 6 | eqtr4d 2859 | . . . 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 225 | . 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑)) |
11 | 1, 10 | sbcied 3813 | 1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Vcvv 3494 [wsbc 3771 ‘cfv 6354 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 |
This theorem is referenced by: (None) |
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