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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 3499 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | spcimgfi1.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3495 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | exim 1831 | . . . 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 2228 | . . 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 1535 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: (None) |
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