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 32301. 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 2840 | . . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
3 | 1, 2 | sylan2b 595 | . . 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 3494 | . . . . . . . 8 ⊢ 𝑝 ∈ V | |
7 | 1, 5, 6 | bnj523 32178 | . . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
8 | 7, 1 | bitr4i 280 | . . . . . 6 ⊢ (𝜑′ ↔ 𝜑) |
9 | 8 | sbcbii 3824 | . . . . 5 ⊢ ([𝐺 / 𝑓]𝜑′ ↔ [𝐺 / 𝑓]𝜑) |
10 | 4, 9 | bitri 277 | . . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
11 | bnj934.50 | . . . 4 ⊢ 𝐺 ∈ V | |
12 | 1, 10, 11 | bnj609 32208 | . . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
13 | 3, 12 | sylibr 236 | . 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″) |
14 | 13 | ancoms 461 | 1 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 [wsbc 3768 ∅c0 4284 ‘cfv 6348 predc-bnj14 31977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-sbc 3769 df-in 3936 df-ss 3945 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 |
This theorem is referenced by: bnj929 32227 |
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