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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj934 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 35003. 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 |
---|---|
bnj934.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj934.4 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) |
bnj934.7 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) |
bnj934.50 | ⊢ 𝐺 ∈ V |
Ref | Expression |
---|---|
bnj934 | ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj934.1 | . . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
2 | eqtr 2758 | . . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
3 | 1, 2 | sylan2b 594 | . . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
4 | bnj934.7 | . . . . 5 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | |
5 | bnj934.4 | . . . . . . . 8 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | |
6 | vex 3482 | . . . . . . . 8 ⊢ 𝑝 ∈ V | |
7 | 1, 5, 6 | bnj523 34880 | . . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
8 | 7, 1 | bitr4i 278 | . . . . . 6 ⊢ (𝜑′ ↔ 𝜑) |
9 | 8 | sbcbii 3852 | . . . . 5 ⊢ ([𝐺 / 𝑓]𝜑′ ↔ [𝐺 / 𝑓]𝜑) |
10 | 4, 9 | bitri 275 | . . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
11 | bnj934.50 | . . . 4 ⊢ 𝐺 ∈ V | |
12 | 1, 10, 11 | bnj609 34910 | . . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
13 | 3, 12 | sylibr 234 | . 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″) |
14 | 13 | ancoms 458 | 1 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = 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-v 3480 df-sbc 3792 df-ss 3980 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: bnj929 34929 |
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