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Mathbox for Andrew Salmon |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fveqsb | Structured version Visualization version GIF version |
Description: Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Ref | Expression |
---|---|
fveqsb.2 | ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) |
fveqsb.3 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
fveqsb | ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6904 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
2 | fveqsb.3 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | fveqsb.2 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbciegf 3816 | . . 3 ⊢ ((𝐹‘𝐴) ∈ V → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓) |
6 | fvsb 43674 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | |
7 | 5, 6 | bitr3id 285 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2105 ∃!weu 2561 Vcvv 3473 [wsbc 3777 class class class wbr 5148 ‘cfv 6543 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 df-fv 6551 |
This theorem is referenced by: (None) |
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