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Mirrors > Home > MPE Home > Th. List > Mathboxes > subtr | Structured version Visualization version GIF version |
Description: Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
subtr.1 | ⊢ Ⅎ𝑥𝐴 |
subtr.2 | ⊢ Ⅎ𝑥𝐵 |
subtr.3 | ⊢ Ⅎ𝑥𝑌 |
subtr.4 | ⊢ Ⅎ𝑥𝑍 |
subtr.5 | ⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌) |
subtr.6 | ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍) |
Ref | Expression |
---|---|
subtr | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subtr.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | subtr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | subtr.3 | . . . . 5 ⊢ Ⅎ𝑥𝑌 | |
5 | subtr.4 | . . . . 5 ⊢ Ⅎ𝑥𝑍 | |
6 | 4, 5 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑥 𝑌 = 𝑍 |
7 | 3, 6 | nfim 1900 | . . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐵 → 𝑌 = 𝑍) |
8 | eqeq1 2737 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
9 | subtr.5 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌) | |
10 | 9 | eqeq1d 2735 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑋 = 𝑍 ↔ 𝑌 = 𝑍)) |
11 | 8, 10 | imbi12d 345 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 → 𝑋 = 𝑍) ↔ (𝐴 = 𝐵 → 𝑌 = 𝑍))) |
12 | subtr.6 | . . 3 ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍) | |
13 | 1, 7, 11, 12 | vtoclgf 3525 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝑌 = 𝑍)) |
14 | 13 | adantr 482 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3449 |
This theorem is referenced by: (None) |
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