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Mirrors > Home > MPE Home > Th. List > ceqsexv2d | Structured version Visualization version GIF version |
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) |
Ref | Expression |
---|---|
ceqsexv2d.1 | ⊢ 𝐴 ∈ V |
ceqsexv2d.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsexv2d.3 | ⊢ 𝜓 |
Ref | Expression |
---|---|
ceqsexv2d | ⊢ ∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsexv2d.3 | . 2 ⊢ 𝜓 | |
2 | ceqsexv2d.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | ceqsexv2d.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | ceqsexv 3492 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
5 | 4 | biimpri 227 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
6 | exsimpr 1872 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥𝜑) | |
7 | 1, 5, 6 | mp2b 10 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-clel 2815 |
This theorem is referenced by: en0 8890 en0OLD 8891 en0r 8893 ensn1 8894 ensn1OLD 8895 0domg 8977 tz9.1 9598 cplem2 9759 karden 9764 pwmnd 18681 2lgslem1 26664 griedg0prc 28010 1loopgrvd2 28249 bnj150 33261 fnchoice 42998 nfermltl8rev 45683 nfermltl2rev 45684 nfermltlrev 45685 nn0mnd 45862 rrx2xpreen 46554 |
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