Description: Technical lemma for bnj852 . 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 | ||
|---|---|---|---|
| Hypotheses | bnj526.1 | ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) | |
| bnj526.2 | ⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑 ) | ||
| bnj526.3 | ⊢ 𝐺 ∈ V | ||
| Assertion | bnj526 | ⊢ ( 𝜑″ ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj526.1 | ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) | |
| 2 | bnj526.2 | ⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑 ) | |
| 3 | bnj526.3 | ⊢ 𝐺 ∈ V | |
| 4 | 1 | sbcbii | ⊢ ( [ 𝐺 / 𝑓 ] 𝜑 ↔ [ 𝐺 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
| 5 | fveq1 | ⊢ ( 𝑓 = 𝐺 → ( 𝑓 ‘ ∅ ) = ( 𝐺 ‘ ∅ ) ) | |
| 6 | 5 | eqeq1d | ⊢ ( 𝑓 = 𝐺 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
| 7 | 3 6 | sbcie | ⊢ ( [ 𝐺 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
| 8 | 2 4 7 | 3bitri | ⊢ ( 𝜑″ ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |