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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj154 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj153 34727. 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 |
---|---|
bnj154.1 | ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) |
bnj154.2 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Ref | Expression |
---|---|
bnj154 | ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj154.1 | . 2 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) | |
2 | bnj154.2 | . . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
3 | 2 | sbcbii 3837 | . 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
4 | vex 3466 | . . 3 ⊢ 𝑔 ∈ V | |
5 | fveq1 6902 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅)) | |
6 | 5 | eqeq1d 2728 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))) |
7 | 4, 6 | sbcie 3820 | . 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
8 | 1, 3, 7 | 3bitri 296 | 1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 [wsbc 3776 ∅c0 4325 ‘cfv 6556 predc-bnj14 34535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-sbc 3777 df-ss 3964 df-uni 4916 df-br 5156 df-iota 6508 df-fv 6564 |
This theorem is referenced by: bnj153 34727 bnj580 34760 bnj607 34763 |
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