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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 6685 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
2 | fveqsb.3 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | fveqsb.2 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbciegf 3811 | . . 3 ⊢ ((𝐹‘𝐴) ∈ V → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ 𝜓) |
6 | fvsb 40791 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | |
7 | 5, 6 | syl5bbr 287 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2114 ∃!weu 2653 Vcvv 3496 [wsbc 3774 class class class wbr 5068 ‘cfv 6357 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 df-uni 4841 df-iota 6316 df-fv 6365 |
This theorem is referenced by: (None) |
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