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Mirrors > Home > NFE Home > Th. List > elsb1 | GIF version |
Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2104 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
elsb1 | ⊢ ([y / x]x ∈ z ↔ y ∈ z) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . . 3 ⊢ Ⅎx w ∈ z | |
2 | 1 | sbco2 2086 | . 2 ⊢ ([y / x][x / w]w ∈ z ↔ [y / w]w ∈ z) |
3 | nfv 1619 | . . . 4 ⊢ Ⅎw x ∈ z | |
4 | elequ1 1713 | . . . 4 ⊢ (w = x → (w ∈ z ↔ x ∈ z)) | |
5 | 3, 4 | sbie 2038 | . . 3 ⊢ ([x / w]w ∈ z ↔ x ∈ z) |
6 | 5 | sbbii 1653 | . 2 ⊢ ([y / x][x / w]w ∈ z ↔ [y / x]x ∈ z) |
7 | nfv 1619 | . . 3 ⊢ Ⅎw y ∈ z | |
8 | elequ1 1713 | . . 3 ⊢ (w = y → (w ∈ z ↔ y ∈ z)) | |
9 | 7, 8 | sbie 2038 | . 2 ⊢ ([y / w]w ∈ z ↔ y ∈ z) |
10 | 2, 6, 9 | 3bitr3i 266 | 1 ⊢ ([y / x]x ∈ z ↔ y ∈ z) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 [wsb 1648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 |
This theorem is referenced by: cvjust 2348 |
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