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Mirrors > Home > MPE Home > Th. List > ceqsalgALT | Structured version Visualization version GIF version |
Description: Alternate proof of ceqsalg 3476, not using ceqsalt 3474. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsalgALT | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3452 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | nfa1 2152 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑) | |
3 | ceqsalg.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | ceqsalg.2 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 232 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 5 | a2i 14 | . . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
7 | 6 | sps 2182 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
8 | 2, 3, 7 | exlimd 2216 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
9 | 1, 8 | syl5com 31 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)) |
10 | 4 | biimprcd 253 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 3, 10 | alrimi 2211 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
12 | 9, 11 | impbid1 228 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: (None) |
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