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| Mirrors > Home > MPE Home > Th. List > spcimgfi1OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of spcimgfi1 3547 as of 27-Jul-2025. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| spcimgfi1.1 | ⊢ Ⅎ𝑥𝜓 | 
| spcimgfi1.2 | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| spcimgfi1OLD | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
| 2 | spcimgfi1.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | issetf 3497 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | 
| 4 | exim 1834 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓))) | |
| 5 | 3, 4 | biimtrid 242 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓))) | 
| 6 | spcimgfi1.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 7 | 6 | 19.36 2230 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) | 
| 8 | 5, 7 | imbitrdi 251 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓))) | 
| 9 | 1, 8 | syl5 34 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 | 
| This theorem is referenced by: (None) | 
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