![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj523 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 31765. 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 3752 | . 2 ⊢ ([𝑀 / 𝑛]𝜑 ↔ [𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
4 | bnj523.3 | . . 3 ⊢ 𝑀 ∈ V | |
5 | 4 | bnj525 31582 | . 2 ⊢ ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
6 | 1, 3, 5 | 3bitri 298 | 1 ⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1520 ∈ wcel 2079 Vcvv 3432 [wsbc 3701 ∅c0 4206 ‘cfv 6217 predc-bnj14 31531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-sbc 3702 |
This theorem is referenced by: bnj600 31763 bnj908 31775 bnj934 31779 |
Copyright terms: Public domain | W3C validator |