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| Description: A closed version of spcimgf 3550. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 27-Jul-2025.) | 
| Ref | Expression | 
|---|---|
| spcimgfi1.1 | ⊢ Ⅎ𝑥𝜓 | 
| spcimgfi1.2 | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| spcimgfi1 | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | spcimgfi1.2 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 2 | spcimgfi1.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | spcimgft 3546 | . . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | |
| 4 | 3 | ex 412 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))) | 
| 5 | 1, 2, 4 | mp2an 692 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 | 
| 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-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 | 
| This theorem is referenced by: spcgft 3549 spcimgf 3550 ss2iundf 43672 spcdvw 49198 | 
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