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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj523 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 34914. 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 |
---|---|
bnj523.1 | ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj523.2 | ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) |
bnj523.3 | ⊢ 𝑀 ∈ V |
Ref | Expression |
---|---|
bnj523 | ⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj523.2 | . 2 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) | |
2 | bnj523.1 | . . 3 ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
3 | 2 | sbcbii 3852 | . 2 ⊢ ([𝑀 / 𝑛]𝜑 ↔ [𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
4 | bnj523.3 | . . 3 ⊢ 𝑀 ∈ V | |
5 | 4 | bnj525 34731 | . 2 ⊢ ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
6 | 1, 3, 5 | 3bitri 297 | 1 ⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 [wsbc 3791 ∅c0 4339 ‘cfv 6563 predc-bnj14 34681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: bnj600 34912 bnj908 34924 bnj934 34928 |
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