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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj526 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 34897. 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 |
---|---|
bnj526.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj526.2 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
bnj526.3 | ⊢ 𝐺 ∈ V |
Ref | Expression |
---|---|
bnj526 | ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj526.2 | . 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) | |
2 | bnj526.1 | . . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
3 | 2 | sbcbii 3865 | . 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ [𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
4 | bnj526.3 | . . 3 ⊢ 𝐺 ∈ V | |
5 | fveq1 6919 | . . . 4 ⊢ (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅)) | |
6 | 5 | eqeq1d 2742 | . . 3 ⊢ (𝑓 = 𝐺 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))) |
7 | 4, 6 | sbcie 3848 | . 2 ⊢ ([𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
8 | 1, 3, 7 | 3bitri 297 | 1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2108 Vcvv 3488 [wsbc 3804 ∅c0 4352 ‘cfv 6573 predc-bnj14 34664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-v 3490 df-sbc 3805 df-ss 3993 df-uni 4932 df-br 5167 df-iota 6525 df-fv 6581 |
This theorem is referenced by: bnj607 34892 |
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