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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 6113 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
2 | fveqsb.3 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | fveqsb.2 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbciegf 3434 | . . 3 ⊢ ((𝐹‘𝐴) ∈ V → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓) |
6 | fvsb 37677 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | |
7 | 5, 6 | syl5bbr 273 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 ∃!weu 2458 Vcvv 3173 [wsbc 3402 class class class wbr 4583 ‘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-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 |
This theorem is referenced by: (None) |
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