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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj609 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 31508. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj609.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj609.2 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
bnj609.3 | ⊢ 𝐺 ∈ V |
Ref | Expression |
---|---|
bnj609 | ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj609.2 | . 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) | |
2 | bnj609.3 | . . 3 ⊢ 𝐺 ∈ V | |
3 | dfsbcq 3635 | . . 3 ⊢ (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑)) | |
4 | fveq1 6410 | . . . 4 ⊢ (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅)) | |
5 | 4 | eqeq1d 2801 | . . 3 ⊢ (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))) |
6 | bnj609.1 | . . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
7 | 6 | sbcbii 3689 | . . . 4 ⊢ ([𝑒 / 𝑓]𝜑 ↔ [𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
8 | vex 3388 | . . . . 5 ⊢ 𝑒 ∈ V | |
9 | fveq1 6410 | . . . . . 6 ⊢ (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅)) | |
10 | 9 | eqeq1d 2801 | . . . . 5 ⊢ (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))) |
11 | 8, 10 | sbcie 3668 | . . . 4 ⊢ ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)) |
12 | 7, 11 | bitri 267 | . . 3 ⊢ ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)) |
13 | 2, 3, 5, 12 | vtoclb 3450 | . 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
14 | 1, 13 | bitri 267 | 1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 ∈ wcel 2157 Vcvv 3385 [wsbc 3633 ∅c0 4115 ‘cfv 6101 predc-bnj14 31274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rex 3095 df-v 3387 df-sbc 3634 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 |
This theorem is referenced by: bnj600 31506 bnj908 31518 bnj934 31522 |
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