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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 2981 | . . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | subtr2.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | subtr2.4 | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
6 | 4, 5 | nfbi 2006 | . . . 4 ⊢ Ⅎ𝑥(𝜓 ↔ 𝜒) |
7 | 3, 6 | nfim 1999 | . . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐵 → (𝜓 ↔ 𝜒)) |
8 | eqeq1 2829 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
9 | subtr2.5 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
10 | 9 | bibi1d 335 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))) |
11 | 8, 10 | imbi12d 336 | . . 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 3480 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) |
14 | 13 | adantr 474 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 Ⅎwnfc 2956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 |
This theorem is referenced by: (None) |
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