Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj154 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj153 32051. 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 3826 | . 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
4 | vex 3495 | . . 3 ⊢ 𝑔 ∈ V | |
5 | fveq1 6662 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅)) | |
6 | 5 | eqeq1d 2820 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))) |
7 | 4, 6 | sbcie 3809 | . 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
8 | 1, 3, 7 | 3bitri 298 | 1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 [wsbc 3769 ∅c0 4288 ‘cfv 6348 predc-bnj14 31857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-rex 3141 df-v 3494 df-sbc 3770 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 |
This theorem is referenced by: bnj153 32051 bnj580 32084 bnj607 32087 |
Copyright terms: Public domain | W3C validator |