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Mirrors > Home > MPE Home > Th. List > Mathboxes > subtr2 | Structured version Visualization version GIF version |
Description: Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
subtr.1 | ⊢ Ⅎ𝑥𝐴 |
subtr.2 | ⊢ Ⅎ𝑥𝐵 |
subtr2.3 | ⊢ Ⅎ𝑥𝜓 |
subtr2.4 | ⊢ Ⅎ𝑥𝜒 |
subtr2.5 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
subtr2.6 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
subtr2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subtr.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | subtr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2988 | . . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | subtr2.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | subtr2.4 | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
6 | 4, 5 | nfbi 1895 | . . . 4 ⊢ Ⅎ𝑥(𝜓 ↔ 𝜒) |
7 | 3, 6 | nfim 1888 | . . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐵 → (𝜓 ↔ 𝜒)) |
8 | eqeq1 2822 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
9 | subtr2.5 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
10 | 9 | bibi1d 345 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))) |
11 | 8, 10 | imbi12d 346 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ↔ (𝐴 = 𝐵 → (𝜓 ↔ 𝜒)))) |
12 | subtr2.6 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
13 | 1, 7, 11, 12 | vtoclgf 3563 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) |
14 | 13 | adantr 481 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 |
This theorem is referenced by: (None) |
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