New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eleq2s | GIF version |
Description: Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
eleq2s.1 | ⊢ (A ∈ B → φ) |
eleq2s.2 | ⊢ C = B |
Ref | Expression |
---|---|
eleq2s | ⊢ (A ∈ C → φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2s.2 | . . 3 ⊢ C = B | |
2 | 1 | eleq2i 2417 | . 2 ⊢ (A ∈ C ↔ A ∈ B) |
3 | eleq2s.1 | . 2 ⊢ (A ∈ B → φ) | |
4 | 2, 3 | sylbi 187 | 1 ⊢ (A ∈ C → φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: elxpi 4800 optocl 4838 ecexr 5950 ectocld 5991 ecoptocl 5996 nulnnc 6118 ncprc 6124 elnc 6125 |
Copyright terms: Public domain | W3C validator |