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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj154 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj153 30204. 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 3458 | . 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
4 | vex 3176 | . . 3 ⊢ 𝑔 ∈ V | |
5 | fveq1 6102 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅)) | |
6 | 5 | eqeq1d 2612 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))) |
7 | 4, 6 | sbcie 3437 | . 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
8 | 1, 3, 7 | 3bitri 285 | 1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 [wsbc 3402 ∅c0 3874 ‘cfv 5804 predc-bnj14 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-sbc 3403 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: bnj153 30204 bnj580 30237 bnj607 30240 |
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