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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 31768. 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 3757 | . 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
4 | vex 3440 | . . 3 ⊢ 𝑔 ∈ V | |
5 | fveq1 6537 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅)) | |
6 | 5 | eqeq1d 2797 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))) |
7 | 4, 6 | sbcie 3741 | . 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 1522 [wsbc 3706 ∅c0 4211 ‘cfv 6225 predc-bnj14 31575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-rex 3111 df-v 3439 df-sbc 3707 df-uni 4746 df-br 4963 df-iota 6189 df-fv 6233 |
This theorem is referenced by: bnj153 31768 bnj580 31801 bnj607 31804 |
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